^ 


A  TREATISE 


ON 


^ORDINARY   AND    PARTIAL 


DIFFERENTIAL   EQUATIONS 


BY 


WILLIAM   WOOLSEY   JOHNSON 

Professor  of  Mathematics  at  the  United  States  Naval  Academy 
Annapolis  Maryland 


THIRD    E^PITION 


SECOND    THOUSAND. 


NEW  YORK: 
JOHN     WILEY    &     SONS, 

53   East  Tenth   Street. 
1893. 


Entered  according  to  Act  of  Congress,  in  the  year  1889,  by 

WILLIAM  WOOLSEY  JOHNSON, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


PREFACE. 


The  treatment  of  the  subject  of  Differential  Equations 
here  presented  will,  it  is  hoped,  be  found  complete  in  all  those 
portions  which  bear  upon  their  practical  applications,  and  in 
the  discussion  of  their  theory  so  far  as  it  can  be  adequately 
treated  without  the  use  of  the  complex  variable.  The  topics 
included  and  the  order  pursued  are  sufficiently  indicated  by 
the  table  of  contents. 

An  amount  of  space  somewhat  greater  than  usual  has  been 
devoted  to  the  geometrical  illustrations  which  arise  when  the 
variables  are  regarded  as  the  rectangular  coordinates  of  a  point. 
This  has  been  done  in  the  belief  that  the  conceptions  pecuHar 
to  the  subject  are  more  readily  grasped  when  embodied  in 
their  geometric  representations.  In  this  connection  the  sub- 
ject of  singular  solutions  of  ordinary  differential  equations  and 
the  conception  of  the  characteristic  in  partial  differential  equa- 
tions may  be  particularly  mentioned. 

Particular  attention  has  been  paid  to  the  development  of 
symbolic  methods,  especially  in  connection  with  the  operator 

X — ,  for  which,  in  accordance  with  recent  usage,  the  symbol  i9 
ax 

has  been  adopted.     Some  new  applications  of  this  symbol  have 
been  made. 


242511 


IV  PREFACE. 

The  expression  "binomial  equations"  is  applied  in  this 
work  (in  a  sense  introduced  by  Boole)  to  those  linear  equations 
which  are  included  in  the  general  form  f^{ff)y  j^ x%{f^)y  =.0^ 
and  which  constitute  the  class  of  equations  best  adapted  to 
solution  by  development  in  series.  In  the  sections  treating 
of  this  method  a  uniform  process  has  been  adopted  for  the 
secondary  or  logarithmic  solutions  which  occur  in  certain  cases. 
The  development  of  the  particular  integral  when  the  second 
member  is  a  power  of  x  is  also  considered.  Chapter  VIII  is 
devoted  to  the  general  solution  of  the  binomial  equation  in 
the  notation  of  the  hypergeometric  series,  and  Chapter  IX  to 
Riccati's,  Bessel's  and  Legendre's  equations. 

The  examples  at  the  ends  of  the  sections  have  been  derived 
from  various  sources,  and  not  a  few  prepared  expressly  for  this 
work.  They  are  arranged  in  order  of  difficulty,  and  the  solu- 
tions are  given.  These  have  been  verified  in  the  proof-sheets, 
so  that  it  is  believed  that  they  will  be  found  free  from  errors. 

The  ordinary  references  in  the  text  are  to  Rice  and  John- 
son's Diff.  Calc.  and  Johnson's  Int.  Calc,  published  by  John 
Wiley  and  Sons  uniformly  with  the  present  volume. 

W.  W.J. 
U.  S.  Naval  Academy, 
May,  1889. 


CONTENTS, 


CHAPTER   I. 

NATURE    AND    MEANING    OF    A   DIFFERENTIAL   EQUATION    BETWEEN   TWO 

VARIABLES. 

I. 

PAGE 

Solutions  in  the  Form  y  =  Fix) i 

Solutions  not  in  the  Form  y  —  Fix) , 3 

Particular  and  Complete  Integrals 3 

Primitive  of  a  Differential  Equation 5 

Number  of  Arbitrary  Constants 6 

Geometrical  Illustration  of  the  Meaning  of  a  Differential  Equation 8 

Systems  of  Curves  containing  an  Arbitrary  Parameter 9 

Doubly  Infinite  Systems  of  Curves 10 

Examples  1 12 


CHAPTER   11. 

EQUATIONS    OF   THE    FIRST   ORDER   AND   DEGREE. 

II. 

Separation  of  the  Variables 14 

Reduction  of  the  Integral  to  Algebraic  Form 15 

Homogeneous  Equations 16 

Similar  and  Similarly  situated  Systems  of  Curves 18 

Case  in  which  the  Coefficients  of  dx  and  dy  are  of  the  First  Degree 19 

Examples  II 20 

III. 

Exact  Differential  Equations 22 

Integrating  Factors 24 


Vi  CONTENTS. 


PAGB 

Expressions  of  the  Form  x<^y^(jnydx  +  nxdy) 27 

Examples  III 29 

IV. 

The  Linear  Equation  of  the  First  Order 30 

Transformation  of  a  Differential  Equation 32 

Extension  of  the  Linear  Equation 33 

Examples  IV 34 


CHAPTER   III. 

EQUATIONS   OF  THE    FIRST   ORDER,    BUT   NOT   OF  THE   FIRST  DEGREE. 

V. 

Decomposable  Equations 37 

Equations  Properly  of  the  Second  Degree 38 

Systems  of  Curves  corresponding  to  Equations  of  Different  Degrees 39 

Standard  Form  of  the  Integral  of  an  Equation  of  the  Second  Degree 42 

Singular  Solutions 43 

The  Discriminant 45 

Cusp-Loci 46 

Tac-Loci  and  Node-Loci 47 

Examples  V 50 

VI. 

Solution  by  Differentiation 52 

Equations  from  which  One  of  the  Variables  is  Absent 54 

Homogeneous  Equations  not  of  the  First  Degree  in  / 57 

The  Equation  of  the  First  Degree  in  x  and  ^ 58 

Clairaut's  Equation 59 

Examples  VI 61 

VII. 

Geometrical  Applications 63 

Polar  Coordinates 64 

The  required  Curve  a  Singular  Solution 66 

Orthogonal  Trajectories 67 

Examples  VII 69 


CONTENTS.  vii 


CHAPTER   IV. 

EQUATIONS  OF  THE   SECOND   ORDER. 

VIII. 

Successive  Integration 72 

The  First  Integrals 74 

Integrating  Factors  .y. 76 

Singular  Solutions  of  Equations  of  the  Second  Order  (foot-note) 77 

Derivation  of  the  Complete  Integral  from  Two  First  Integrals 78 

Exact  Equations  of  the  Second  Order 80 

Equations  in  which  y  does  not  occur 82 

Equations  in  which  x  does  not  occur yj, 83 

The  Method  of  Variation  of  Parameters 84 

Examples  VIII 87 

CHAPTER   V. 

LINEAR   EQUATIONS  WITH   CONSTANT   COEFFICIENTS. 

IX. 

Properties  of  the  Linear  Equation 91 

The  Linear  Equation  with  Constant  Coefficients  and  Second  Member  Zero. ...  93 

Case  of  Equal  Roots 95 

Case  of  Imaginary  Roots  97 

The  Linear  Equation  with  Constant  Coefficients  and  Second  Member  a  Func- 
tion oi  X 98 

The  Inverse  Operative  Symbol 99 

General  Expression  for  the  Integral loi 

Examples  IX 103 

X. 

Symbolic  Methods  of  Integration 106 

The  Second  Member  X  of  the  Form  e^x 106 

Case  in  which  X  contains  a  Term  of  the  Form  %\w  ax  ox  co^  ax 109 

Case  in  which  X  contains  Terms  of  the  Form  x*n 112 

Symbolic  Formula  of  Reduction  for  the  Form  e^x  V. 114 

Application  to  the  Evaluation  of  an  Ordinary  Integral  (see  also  p.  1 18) 116 

Symbolic  Formula  of  Reduction  for  the  Form  xV. 116 

Symbolic  Formula  of  Reduction  for  the  Form  xrV 118 

Employment  of  the  Exponential  Forms  of  sin  ax  and  cos  ajr 119 

Examples  X 120 


Vlll  CONTENTS. 


CHAPTER  VI 

LINEAR   EQUATIONS  WITH   VARUBLE   COEFnCIENTS. 
XI. 


PAGE 


The  Homogeneous  Linear  Equation 12^ 

The  Operative  Symbol  tV 

Complete  Integral  of  the  Equation  f(^H)y  =  o 127 

Cases  of  Equal  and  Imaginary  Roots 127 

The  Particular  Integral  oif{fi)y  -X .!..!!!  128 

Case  in  which  X  is  of  the  Form  x<* ,2g 

Symbol  Solutions  of  Linear  Equations  with  Variable  Coefficients 130 

Non-Commutative  Symbohc  Factors j  ^2 

Examples  XI . _^ 

'  ■ " *34 

XII. 

Exact  Linear  Equations j ^c 

The  Condition  of  Direct  Integrability i  ^y 

^  Integrating  Factors  of  the  Form  x^ ,  ^o 

Symbolic  Treatment  of  Exact  Linear  Equation 140 

Symbolic  Formulae  involving  D  and  »? ,41 

Examples  XII j 

XIII. 

The  Linear  Equation  of  the  Second  Order 14- 

Case  in  which  an  Integral  ji,  when  the  Second  Member  is  Zero,  is  known 147 

Expression  for  the  Complete  Integral  in  Terms  of  jf/j 149 

Relation  between  Two  Independent  Integrals  ji  and^/a 151 

Symmetrical  Expression  for  the  Particular  Integral i  r2 

Resolution  of  the  Operator  into  Factors ic^ 

The  Related  Equation  of  the  First  Order i  r^ 

The  Transformation  y  =  v/(x)    jc- 

The  Transformation  y  =  eo^"*v jcy 

Removal  of  the  Term  containing  the  First  Derivative  —  the  Normal  Form  ....  158 

The  Invariant  for  the  Transformation  y  =  v/(x)  (see  also  Ex.  26,  p.  165) 159 

Change  of  the  Independent  Variable i5o 

Examples  XIII j52 


CONTENTS.  ix 


CHAPTER  VII. 

SOLUTIONS   IN   SERIES. 
^  XIV. 

PAGE 

Development  of  the  Integral  of  a  Differential  Equation  in  Series i66 

Development  of  the  Independent  Integrals  of  a  Linear  Equation  whose  Second 

Member  is  Zero 167 

Convergency  of  the  Series 171 

Development  of  the  Particular  Integral 172 

Binomial  and  Polynomial  Equations 1 73 

Finite  Solutions 1 74 

Examples  XIV 177 

XV. 

Development  of  the  Logarithmic  Form  of  the  Second  Integral  —  Case  of  Equal 

Values  of  w 181 

Case  in  which  the  Values  of  m  differ  by  a  Multiple  of  j 185 

Special  Forms  of  the  Particular  Integral 191 

Examples  XV 194 

CHAPTER   VIIL 

THE   HYPERGEOMETRIC   SERIES. 

XVL 

General  Solution  of  the  Binomial  Equation  of  the  Second  Order 198 

Differential  Equation  of  the  Hypergeometric  Series 201 

Integral  Values  of  7  and  7' 202 

The  Supplementary  Series  when  y^,  is  a  Finite  Series 204 

Imaginary  Values  of  o  and  fi 205 

Infinite  Values  of  a  and  ^ 206 

Case  in  which  a  or  j8  equals  7  or  Unity 208 

The  Binomial  Equation  of  the  Third  Order 209 

Development  of  the  Solution  in  Descending  Series   210 

Transformation  of  the  Equation  of  the  Hypergeometric  Series 211 

Change  of  the  Independent  Variable 214 

The  Twenty-Four  Integrals 216 

Solutions  in  Finite  Form 218 

Examples  XVI 220 


CONTENTS. 


>/. 


CHAPTER  IX. 

SPECIAL   FORMS   OF   DIFFERENTIAL   EQUATIONS. 
XVII. 


PAGE 

Riccati's  Equation 224 

Standard  I^inear  Form  of  the  Equation 225 

Finite  Solutions 228 

Relations  between  the  Six  Integrals 230 

Transformations  of  Riccati's  Equation 232 

Bessel's  Equation 234 

Finite  Solutions 235 

The  Besselian  Functions •  • . .  237 

The  Besselian  Functions  of  the  Second  Kind 239 

Legendie's  Equation 241 

The  Legendrean  Coefficients 243 

The  Second  Integral  Qn  when  n  is  an  Integer 244 

Examples  XVII 247 


CHAPTER   X. 

EQUATIONS   INVOLVING   MORE   THAN   TWO   VARIABLES. 

XVIII. 

Determinate  Systems  of  the  First  Order 25 1 

Transformation  of  Variables 252 

Exact  Equations 253 

The  Integrals  of  a  System 254 

Equations  of  Higher  Orders  equivalent  to  Determinate  Systems  of  the  First 

Order 256 

Geometrical  Meaning  of  a  System  involving  Three  Variables 257 

Examples  XVIII 258 

XIX. 

Simultaneous  Linear  Equations 260 

Number  of  Arbitrary  Constants 263 

Introduction  of  a  New  Variable 264 

Examples  XIX 266 


CONTENTS.  xi 


XX. 

PAGE 

Single  Differential  Equations  involving  more  than  Two  Variables 270 

The  Condition  of  Integrability 270 

Solution  of  the  Integrable  Equation 272 

Separation  of  the  Variables 274 

Homogeneous  Equations 275 

Equations  containing  more  than  Three  Variables 276 

The  Non-Integrable  Equation 278 

Monge's  Solution - 280 

Geometrical  Meaning  of  a  Single  Differential  Equation  between  Three  Variables  281 

The  Auxiliary  System  of  Lines 282 

Distinction  between  the  Two  Cases 283 

Examples  XX 284 


CHAPTER   XI. 

PARTIAL   DIFFERENTIAL   EQUATIONS   OF   THE    FIRST   ORDER. 

XXI. 

Equations  involving  a  Single  Partial  Derivative 287 

Equations  of  the  First  Order  and  Degree 288 

Lagrange's  Solution 289 

Geometrical  Illustration  of  Lagrange's  Solution 291 

Orthogonal  Surfaces 292 

The  Complete  and  General  Primitives 293 

Derivation  of  the  Differential  Equation  from  the  General  Primitive 294' 

Examples  XXI 297 

XXII. 

The  Non-Linear  Equation  of  the  First  Order 299 

The  System  of  Characteristics 3°° 

The  General  Integral 3^4 

Derivation  of  a  Complete  Integral  from  the  Equations  of  the  Characteristic 306 

Relation  of  the  General  to  the  Complete  Integral 309 

Singular  Solutions 3  ^  ^ 

Integrals  having  Single  Points  of  Contact  with  the  Singular  Solution 312 

Derivation  of  the  Singular  Solution  from  the  Differential  Equation 313 

Equations  involving  p  and  q  only 3^4 

Integrals  formed  by  Characteristics  passing  through  a  Common  Point 314 

Equation  Analogous  to  Clairaut's 3^5 

Equations  not  containing  x  ox  y ". 3^^ 


XU  CONTENTS. 


PAGB 


Equations  of  the  Form  fi{x,p)  =f-i{y,  q) 317 

Change  of  Form  in  the  Equations  of  the  Characteristic 319 

Transformation  of  the  Variables 321 

Examples  XXII 323 


CHAPTER  XXL 

PARTIAL  DIFFERENTIAL   EQUATIONS   OF   HIGHER   ORDER. 

xxiir. 

Equations  of  the  Second  Order 326 

The  Primitive  containing  Two  Arbitrary  Functions 326 

Forms  which  give  Rise  to  Equations  of  the  Second  Order 327 

The  Intermediate  Equation  of  the  First  Order 329 

Successive  Integration 330 

Monge's  Method 332 

Integrability  of  Monge's  Equations ■^■^z^ 

Illustrative  Examples 33  e 

Examples  XXIII 340 

XXIV. 

Linear  Equations 341 

>mogeneous  Equations  with  Constant  Coefficients 342 

^  .nbolic  Solution  of  the  Component  Equations  of  the  Form  (/)  —  mD')z  =  o. .  344 

Case  of  Equal  Roots 34r 

Case  of  Imaginary  Roots   347 

The  Particular  Integral 348 

The  Second  Member  of  the  Form  4»(a;r  +  by) 350 

The  Non-Homogeneous  Equation 3C2 

Special  Forms  of  the  Integral 354 

Special  Methods  for  the  Particular  Integral 355 

The  Second  Member  of  the  Form  e"^  +  h 3^6 

The  Second  Member  of  the  Form  sm{ax  +  by)  or  cos{ax  -{•  by) 357 

The  Second  Member  of  the  Form  x^y^ 358 

The  Second  Member  of  the  Form  e^^  +  hy 360 

Linear  Equations  with  Variable  Coefficients 361 

The  Equation  F{fy,  fy')z  =  0 , 363 

The  Equation  eIj%  //'>  =V 364 

The  Symbol  »9  +  ^9'  in  Relation  to  the  Homogeneous  Function  of  x  and  y  . . .  .  365 

Examples  XXIV 366 


DIFFERENTIAL   EQUATIONS, 


CHAPTER  I. 

NATURE  AND    MEANING   OF  A  DIFFERENTIAL   EQUATION   BETWEEN 
TWO    VARIABLES. 

I. 

Solutions  in  the  Form  y  =  F{x). 

I.  In  the  Integral  Calculus,  we  suppose  the  differential  of  a 
variable  y  to  be  given  in  terms  of  another  variable  x  and  ii  ' 
differential  dx,  and  we  seek  to  express  y  as  a  function  of  x ; 
in  other  words,  since  we  know  that  the  form  of  the  given 
equation  must  be 

dy  =  f{x)dx,  ........    (i) 

which  may  be  written 

l  =  /(^>' <^> 

the  derivative  of  j/  is  given  in  terms  of  x^  and  an  equation 
of  the  form 

J  =  ^W (3) 

is  said  to  satisfy  the  given  equation  (i)  or  (2)  when  F{x)  is  a 
function  whose  derivative  is  the  given  function  f{x). 

I 


2  DIFFERENTIAL  EQUATIONS.  [Art.  2. 

2.  A  differential  equation  between  two  variables  x  and  y 
is  an  equation  involving  in  any  manner  one  or  more  of  the 
derivatives  of  the  unknown  function  y  with  respect  to  Xy 
together  with  one  or  both  of  the  variables  x  and  y.  The 
order  of  the  equation  is  that  of  the  highest  derivative  contained 
in  it,  and  its  degree  is  that  of  the  highest  power  of  this  deriva- 
tive which  occurs.  An  equation  of  the  form  y  =  F{x)  satisfies 
the  differential  equation  if  the  substitution  of  F{x)  and  its 
derivatives  for  y  and  its  derivatives  reduces  it  to  an  identity. 
For  example,  the  differential  equation 

dx^         dx 

will  be  found  on  trial  to  be  satisfied  by  j/  =  ^-^  sin  x ;  for,  if  we 
substitute  this  value  for  j,  and  for  its  derivatives  the  resulting 

values  -^  =■  ^(cos  x  4-  sin  x)  and   — ^  =  2e^  cos  x,  the  first 
dx  dx^ 

member  reduces  to  zero. 

3.  Equation  (i)  is,  in  fact,  the  simplest  form  of  differential 
equation.     Its  general. solution  is  expressed  by  the  formula 


=  f/(^ 


)dx', (4> 


and  it  is  the  province  of  the  Integral  Calculus  to  reduce  this 
expression,  when  possible,  to  a  form  free  from  the  integral  sign, 
and  involving  only  known  functional  symbols.  But,  when  this 
is  not  possible,  the  second  member  of  equation  (4)  represents  a 
new  functional  form,  which,  by  definition,  satisfies  equation  (i) ; 
so  that  the  formula  is  still  the  solution  of  the  differential  equa- 
tion. In  like  manner,  a  differential  equation  of  any  other  form 
is  said  to  be  solved  when  a  proper  expression  is  found,  even 
though  it  involve  integrals  which  we  are  unable  to  reduce. 


§  I.]  PARTICULAR  AND    COMPLETE  INTEGRALS. 


Solutions  not  in  the  Form  y  =  F{oc). 

4.  A  relation  between  x  and  y  not  in  the  form  y  =.  f{x) 
may  satisfy  a  differential  equation.  When  this  is  the  case,  the 
values  of  the  derivatives  employed  in  verifying  will  be  expressed 
in  terms  of  x  and  y ;  and,  when  these  are  substituted  in  the 
differential  equation,  the  result  is  a  relation  between  x  and  y 
which  should  be  true  in  virtue  of  the  integral  equation.  For 
example,  in  order  to  show  that  the  equation 

KIT-^l+^=° ('> 

is  satisfied  by 

f  =  4^^, (2) 

we  differentiate  equation  (2) ;  thus, 

and,  substituting  the  value  of  -^  from  (3)  in  equation  (i),  we 

Q/X 

have 

f 


4^^ 
X- 2^  +  ^  =  o* 


This  equation  is  true  by  virtue  of  the  integral  relation  (2) ; 
equation  (2)  is  therefore  a  solution  of  the  given  differential 
equation  (i). 

Particular  and  Complete  Integrals. 

5.  If  Fix)  is  a  particular  value  of  the  integral  in  equation 
(4),  Art.  3,  then 

y  =  F{x)  +  C, 


4  DIFFERENTIAL   EQUATIONS.  [Art.  5. 

where  6^  is  a  constant  to  which  any  value  may  be  assigned,  is 
the  general  or  complete  solution  of  equation  (i).  Thus,  the 
general  solution  involves  an  arbitrary  constant  which  is  called 
the  constant  of  integration.  In  like  manner,  a  solution  of  any 
differential  equation  is  called  a  particular  integral  of  the  equa- 
tion ;  but  the  most  general  solution,  which  is  called  the  complete 
integral,  contains  one  or  more  arbitrary  constants  of  integration, 
the  manner  in  which  these  constants  enter  the  equation  depend- 
ing on  the  form  of  the  differential  equation. 

For  example,  it  was  noticed  in  Art.  2  that  the  differential 
equation 

jy      2-^  +  2y  =  o (i) 

ax^         ax 

is  satisfied  by 

y  ■=  e'^smx, (2) 

which  is,  therefore,  a  particular  integral.  It  is  not  difficult,  in 
this  case,  to  infer  from  this  solution  the  complete  integral ; 
for,  in  the  first  place,  it  is  evident  that,  if  we  multiply  the 
value  of  y  given  in  equation  (2)  by  the  constant  C,  the  values 

of  ^  and  ^-^  will  also  be  multiplied  by  C,  so  that  the  result  of 
dx  dx^ 

substitution  in  the  first  member  of  equation  (i)  will  be  C  times 

the  previous  result,  and  therefore  still  equal  to  zero.     Thus, 

y  =  C^^sin.^ (3) 

is  a  more  general  solution  of  the  differential  equation.  Again, 
since  x  does  not  explicitly  enter  equation  (i),  and  -r  +  a,  where 
a  is  a  constant,  has  the  same  differential  as  x, 

^  =  CV^ +  «  sin  (.:*:  +  a) (4) 

satisfies  the  equation,  and  forms  a  still  more  general  solution. 


§  I.]  PRIMITIVE   OF  A    DIFFERENTIAL   EQUATION.  5 

Expanding  sin  i^x  +  a),  and  putting 

CV^cosa  =  A^     C^^sina  =  By 

we  may  write  the  solution  in  the  form 

y  =  Ae^^YCix  4-  Be^CQ^x, (5) 

in  which  A  and  B  are  two  independent  arbitrary  constants, 
because  C  and  a  are  independently  arbitrary.  We  shall  see 
presently  that  this  equation  containing  two  arbitrary  constants 
is  the  complete  integral  of  equation  (i).  The  particular  integral 
(2)  is  the  result  of  putting  A  =  i  and  ^  =  o  in  the  complete 
integral. 

Primitive  of  a  Differential  Equation. 

6.  The  general  solution  found  in  the  preceding  article  may, 
of  course,  be  verified  by  the  substitution  of  the  values  of  j^, 

-^,  and  -^  in  the  differential  equation.     Thus,  from 
dx  dx^  ^ 

y  =  Ae^sinx  4-  ^^^cos^,  ..........     (i) 

we  get 

-i'  =  Ae^(smx  +  cos^)  +  Be^{cosx  —  sin;\?),.     .     (2) 
ax 

and 

-— ■=■  2Ae^Q.o's>x  —  2Be^smx; (3) 

dx^ 

andy  if  these  values  are  substituted  in  the  first  member  of 

^        ,f+,y=0,      .....      .      (4) 

dx^  ax 

the  coefficients  of  A  and  B  separately  reduce  to  zero,  and  the 
equation  .is  satisfied  independently  of  the  values  of  A  and  B. 


6  DIFFERENTIAL  EQUATIONS,  [Alt.  6. 

It  thus  appears  that  the  differential  equation  (4)  is  the  same  as 
the  result  of  eliminating  A  and  B  from  equations  (i),  (2),  and 
(3).  Equation  (i)  is,  in  this  point  of  view,  said  to  be  the  primi- 
tive of  equation  (4). 

7.  So  also  any  equation  containing  arbitrary  constants  is 
the  primitive  of  a  certain  differential  equation  free  from  those 
constants. 

For  example,  if,  in  the  equation 

c^x  —  cy  -\r  a  —  o, (i) 

c  is  regarded  as  an  arbitrary  constant,  we  have,  by  differentia- 
tion, 

^-^:7  =  o,   or    /  =  <^; (2) 

ax  dx 


whence,  eliminating  c,  we  obtain 


-©■- 


't*'-" « 


as  the  equation  of  which  equation  (i)  is  the  primitive. 

Again,  equation  (2),  from  which  a  has  disappeared  by  differ- 
entiation, is  itself  the  equation  derived  from  equation  (i)  as  a 
primitive,  when  a  is  regarded  as  an  arbitrary,  and  ^  as  a  fixed, 
constant.  But,  if  both  a  and  c  are  arbitrary,  differentiating 
again,  we  have 

dx' 

and,  c  having  disappeared,  this  is  the  equation  of  the  second 
order  of  which  equation  (i)  is  the  primitive. 

.  It  is  evident  that,  in  every  case,  the  number  of  differentia- 
tions necessary,  and  therefore  the  index  of  the  order  of  the 
differential  equation  produced,  will  be  the  same  as  the  number 
of  constants  to  be  eliminated. 


§  L]  NUMBER    OF  ARBITRARY  CONSTANTS.  7 

8.  Considering  now  the  differential  equation  as  given,  the 
primitive  is  an  integral  equation  which  satisfies  it,  the  constants 
eliminated  being,  in  the  reverse  process  of  finding  the  integral, 
the  constants  of  integration  ;  and  it  is  the  most  general  solution, 
or  complete  integral,  because  no  greater  number  of  constants 
could  be  eliminated  without  introducing  derivatives  higher  than 
the  highest  which  occurs  in  the  given  equation.  For  example, 
the  process  given  in  the  preceding  article  shows  that 

c^x  —  cy  -{-  a  =  o 
is  the  complete  integral  of 


^\dx) 


dy    , 
y-f  -^  a  —  o. 
ax 


It  was  shown  in  Art.  4  that  this  differential  equation  is 
satisfied  by  y'^  =  4ax,  which,  it  will  be  noticed,  is  not  a  particu- 
lar case  of  the  complete  integral.  Thus,  while  the  complete 
integral  is  the  most  general  solution,  it.  does  not,  in  all  cases, 
include  all  the  solutions. 

9.  We  thus  see  that  the  complete  integral  of  a  differential 
equation  of  the  first  order  should  contain  one  constant  of 
integration,  that  of  an  equation  of  the  second  order  should 
contain  two  constants,  and  so  on.  It  is,  of  course,  to  be  under- 
stood that  no  two  of  the  constants  admit  of  being  replaced  by 
a  single  one.  For  example,  the  constants  C  and  a  in  the  equa- 
tion 

y  =  C^^  +  " 

are   equivalent   to   a   single   arbitrary   constant ;    for,    putting 
A  =  CV",  the  equation  may  be  written 

y  =  Ae^, 

hence  it  is  the  complete  integral  of  an  equation  of  the  first, 
not  of  the  second,  order. 


DIFFERENTIAL   EQUATIONS.  [Art.   10. 


Geometrical  Illustration  of  the  Meaning  of  a  Differential  Equation. 

10.    Let  X  and  ^  in  a  differential  equation  be  regarded  as 
the  rectangular  coordinates  of  a  point  in  a  plane ;   then  the 

derivative  -^  is  the  tangent  of  the  inclination  to  the  axis  of  x 
dx 

of  the  direction  in  which  the  point  (jir,  y)  is  moving.     Putting 


/  = 


^dy 


dx' 


a  differential  equation  of  the  first  order  is  a  relation  between  the 
variables  Xy  y^  and  /,  of  which  x  and  y  determine  the  position 
of  the  point,  and  /  the  direction  of  its  motion.  We  may  assign 
to  X  and  y  any  values  we  choose,  and  then  determine  from  the 
equation  one  or  more  values  of  /.  We  cannot,  therefore,  regard 
the  differential  equation  as  satisfied  by  certain  points  (that  is, 
by  certain  associated  values  of  x  and  y) ;  but  it  is  satisfied  by 
certain  associated  values  of  x,  j/,  and  /,  that  is,  by  a  point  in 
any  position,  provided  it  is  moving  in  the  proper  direction. 

II.  Let  us  now  suppose  the  point  {x,  y)  to  start  from  any 
assumed  initial  position,  and  to  move  in  the  proper  direction. 
We  have  thus  a  moving  point  satisfying  the  differential  equa- 
tion. As  the  point  moves,  the  values  of  x  and  y  vary,  so  that 
the  value  of  /  derived  from  the  equation  will  likewise,  in  gen- 
eral, vary ;  and  we  may  suppose  the  direction  of  the  point's 
motion  to  vary  in  such  a  way  that  the  moving  point  continues 
to  satisfy  the  differential  equation.  The  line  which  the  pomt 
now  describes  is,  in  general,  a  curve  ;  and  the  point  may  evi- 
dently move  along  this  curve  in  either  direction,  and  yet  always 
satisfy  the  differential  equation.  The  moving  point  may  return 
to  its  initial  position,  thus  describing  a  closed  curve  ;  or  it  may 
pass  to  infinity  in  both  directions,  describing  an  infinite  branch 
of  a  curve. 


§  I.]  GEOMETRICAL   REPRESENTATION.  9 

If,  now,  we  can  determine  the  equation  of  this  curve  in  the 

form  of  an  ordinary  equation  between  x  and  j,  the  value  of  -^ 

dx 

found  by  differentiating  the  equation  of  the  curve  will,  by 
hypothesis,  be  identical,  for  any  values  of  x  and  y^  with  the 
value  of  /  corresponding  to  the  same  values  of  x  and  y  in 
the  differential  equation.  The  equation  of  the  curve  will,  there- 
fore, be  a  solution,  or  integral,  of  the  differential  equation. 

12.  But,  since  this  integral  equation  restricts  the  point  to 
certain  positions,  the  assemblage  of  which  constitutes  the 
curve,  it  is  not  the  complete  solution  of  the  differential  equa- 
tion ;  for  the  complete  solution  ought  to  represent  the  moving 
point  satisfying  the  equation  in  all  its  possible  positions.  If, 
now,  we  take  for  initial  point  any  point  not  on  the  curve  already 
determined,  and  proceed  in  like  manner,  we  shall  determine 
another  curve,  whose  equation  will  be  another  particular  solu- 
tion, or  integral,  of  the  differential  equation.  We  thus  have  an 
unlimited  number  of  curves  forming  a  system  of  curves,  and  the 
complete  integral  is  the  general  equation  of  this  system. 

This  general  equation  must  contain,  besides  x  and  y,  a. 
quantity  independent  of  x  and  y  called  the  parameter^  by  giving 
different  values  to  which  we  obtain  the  equations  of  all  the 
particular  curves  of  the  system.  The  arbitrary  parameter  of 
the  system  is,  of  course,  the  constant  of  integration. 

13.  We  may  illustrate  this  by  a  simple  example.  Let  the 
differential  equation  be 

.    1"=-^ (■) 

ax  y 

y  , 
Since  —  is  the  tangent  of  the  inclination  to  the  axis  of  x  of  the 

line  joining  the  point  {x,  y)  to  the  origin,  the  equation  expresses 
that  the  point  (x,  y)  is  always  moving  in  a  direction  perpendicu- 
lar to  the  line  joining  it  to  the  origin.    Starting  from  any  initial 


10  DIFFERENTIAL  EQUATIONS.  [Art.   1 3. 

position,  it  is  clear  that  the  point  describes  a  circle  about  the 
origin  as  centre.  The  system  of  curves  in  this  case,  therefore, 
consists  of  all  circles  whose  centres  are  at  the  origin ;  and  the 
general  equation  of  this  system, 

x^  ^  f  ^C, (2) 

where  C  is  the  parameter,  is  the  complete  integral. 

Now  consider  the  moving  point  when  in  any  special  posi- 
tion, as,  for  instance,  in  the  position  (3,  2) ;  we  find,  by  substi- 
tuting these  values  for  x  and  y  in  equation  (2), 

C  =  13. 
Hence 

^^  +  /  =  13 

is  the  equation  of  the  particular  curve  in  which  the  point  is 
then  moving.     If  we  differentiate  this  equation,  we  find  a  value 

for  -^  at  the  point  (3,  2)  identical  with  that  given  for  the  same 
ax 

point  by  equation  (i). 


Doubly  Infinite  Systems  of  Curves. 

14.  In  the  case  of  a  differential  equation  of  the  second  order, 
let 

-^  —  p    and     -^  —  Q'i 

then  the  equation  is  a  relation  between  x,  y,  p,  and  q.  It  is 
possible  to  assign  any  values  we  please  to  x,  y,  and  /,  and  to 
determine  from  the  equation  a  value  of  q,  which,  in  connection 
with  the  assumed  values  of  x,  y,  and  /,  will  satisfy  the  equc 
This  value  of  q,  in  connection  with  the  assumed  value 
determines  the  curvature   of   the   path   of   the  moving 


ection 

\ 


§  I.]  GEOMETRICAL   representation:  II 

(;r,  y).  Hence  a  differential  equation  of  the  second  order  may 
be  regarded  as  satisfied  by  a  moving  point  having  any  assumed 
position,  and  moving  in  any  assumed  direction,  provided  only 
that  its  path  have  the  proper  curvature.  Starting  from  any 
assumed  initial  point,  and  in  any  assumed  initial  direction,  the 
point  {Xy  y)  may  move  in  such  a  manner  as  to  satisfy  the  equa- 
tion. As  it  moves  the  values  of  x  and  y  will  vary ;  and,  since 
the  path  has  a  definite  curvature  at  this  point,  the  value  of  / 
will  likewise  vary.  Hence  the  value  of  q  derived  from  the 
differential  equation  will,  in  general,  also  vary ;  but  we  may 
suppose  the  curvature  of  the  path  to  vary  in  such  a  manner 
that  the  moving  point  continues  to  satisfy  the  equation.  A 
curve  is  thus  described  whose  ordinary  equation  is  a  solution 
of  the  differential  equation,  since  the  simultaneous  values  of  x, 

y,  -^,  and  — =^,  at  every  point  of  it,  by  hypothesis,  satisfy  that 

equation. 

15.  As  before,  the  complete  integral  is  the  general  equation 
of  the  system  of  curves  which  may  be  generated  in  the  manner 
explained  above  ;  but  this  system  has  a  greater  generality  than 
that  which  represents  a  differential  equation  of  the  first  order. 
For,  in  its  general  equation,  it  must  be  possible  to  assign  any 
assumed  simultaneous  values  to  x^  y,  and  /.  Substituting  the 
assumed  values  in  the  general  equation  and  in  the  result  of  its 
differentiation,  we  have  two  equations ;  and,  in  order  to  satisfy 
them,  we  must  have  two  arbitrary  parameters  at  our  disposal. 

The  system  of  curves  representing  a  differential  equation  of 
the  second  order  is,  therefore,  a  system  containing  two  param- 
eters, to  each  of  which  independently  an  unlimited  number  of 
values  may  be  assigned.  Such  a  system  is  said  to  be  a  doubly 
ittfinite  system  of  curves. 

In  like  manner,  it  may  be  shown  that  a  differential  equation 
of  the  third  order  is  represented  by  a  triply  infinite  system  of 
curves,  and  so  on. 


12  DIFFERENTIAL   EQUATIONS.  [Art.   1 5. 


Examples  I. 

1.  Form  the  differential  equation  of  which  >'  =  ^cos  a:  is  the  com- 
plete integral. 

-^  +  y\2Xix  =  o. 
ax 

2.  Form  the  equation  of  which  y  =  ax^  +  bx  is  the  complete  inte- 
gral, a  and  b  being  arbitrary. 

,  d^y  dy    , 

XT — =^  —   2X-^  -f-  2y  =  o. 
dx'  dx 

3.  Form  the  equation  of  which  y^  —  2cx  —  ^r^  =  o  is  the  complete 
integral. 

H-   2X-^  —  y  =  o. 

dx 


it) 


4.  Form  the  equation  of  which  ^^^  -f-  2cxey  -f-  ^^  _  q  jg  ^j^g 
primitive. 

5.  Form  the  equation  of  which  y  =  {x  +  c)e^^  is  the  complete 
integral. 

-^  =  (?«-^  +  ay. 
dx 

> 

6.  Denoting  by  B  the  inclination  to  the  axis  of  x  of  the  line  joining 
{x,  y)  to  the  origin,  and  by  ^  the  inclination  of  the  point's  motion,  write 
the  differential  equation  which  expresses  that  ^  is  the  supplement  of  B, 
and  show  that  it  represents  a  system  of  hyperbolas. 

7.  With  the  same  notation,  write  the  differential  equation  which 
expresses  that  "<^  =  2B,  and  show  that  it  represents  a  system  of  circles 
passing  through  the  origin. 


§  L]  EXAMPLES.  13 

8.  Form  the,  differential  equation  of  the  system  of  straight  lines 
which  touch  the  circle  ^r^  -j^  jv^  =  i,  and  show  that  this  circle  also 
satisfies  the  equation. 

9.  Find  the  differential  equatioh  of  all  the  circles  having  their  radii 
equal  to  a. 

i'.(i)"-j=-(g)' 

10.  Find  the  differential  equation  of  all  the  conies  whose  axes 
coincide  with  the  coordinate  axes. 

•^  dx         \dx)         ^  dx'' 


14  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  1 6. 


THAPTER  II. 

EQUATIONS  OF  THE  FIRST  ORDER  AND  DEGREE. 

11. 

Separation  of  the  Variables, 

i6.  In  an  equation  of  the  first  order,  it  is  immaterial  whether 
X  or  j/hQ  taken  as  the  independent  variable.  If  the  equation 
is  also  of  the  first  degree,  it  is  frequently  written  in  the  form 

Mdx  +  Ndy  —  o, 

in  which  M  and  N  denote  functions  of  x  and  y.  The  simplest 
case  is  that  in  which  the  equation  may  be  so  writter^  that  the 
coefficient  of  dx  is  a  function  of  x  only,  and  that  oi  j/  a,  function 
of  jj/  only ;  in  other  words,  the  case  in  which  the  equation  can 
be  written  in  the  form 

/(x)dx  +  <l>{y)dy  =  o (i) 

The  complete  integral  is  then  evidently 

^/{x)dx+^cl>iy)dy  =  C. (2) 


§  IL]  SEPARATION  OF  THE    VARIABLES.  1 5 

17.  The  process  of  reducing  an  equation,  when  possible,  to 
the  form  (i)  is  called  the  separation  of  the  variables.  For 
example,  in  the  equation 

{\  —  y)dx -\- {\-ir  x)dy  =  o, (i) 

the  variables  are  separated  by  dividing  by  (i  —  /)(i  +  ;tr) ; 
thus, 

-^  +  ^:-  =  o (2) 

\  ->r  X       \  —  y 
Hence,  integrating, 

log(i  +  ^)  -  log(i  --  y):=M  c (3) 

18.  The  integral  here  presents  itself  in  a  transcendental 
form ;  but  it  is  readily  reduced  to  an  algebraic  form,  for  (3) 
may  be  written  in  the  form 

log^-^  =  ^; (4) 

whence 

7^;=-- <=> 

or,  putting  C  for  ^, 

I  +  ^  =  C(i  -  >') (6) 

It  is  to  be  noticed  that  C  in  equation  (6)  admits  of  all  values 
positive  and  negative,  although  e^  can  only  be  positive.  In  fact, 
equation  (4)  is  defective  in  notation ;  for,  since  the  integrals 
are  the  logarithms  of  the  numerical  values  of  i  +  ;ir  and  \  —  y 
respectively  (see  Int.  Calc,  Art.  10),  that  equation  ought  strictly 
to  have  been  written 


and  finally  C  is  put  for  ±^. 


c. 


1 6  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.   1 9. 

19.  The  complete  integral  in  the  above  example  is  readily 
seen  to  be  the  equation  of  a  system  of  straight  lines  passing 
through  the  point  (—1,  i).  In  general,  any  assumed  simulta- 
neous values  of  x  and  y,  that  is,  any  assumed  position  of  the 
moving  point,  determines  a  value  of  Cy  as  in  Art.  13.  But,  for 
the  particular  point  (—1,  i),  the  value  of  C  is  indeterminate  in 

equation  (6) ;  and  accordingly  we  find  that  /  or  -^  is  also  inde- 

dx 

terminate  for  this  point  in  equation  (i). 

It  must  not,  however,  be  assumed  that,  whenever  /  in  the 
differential  equation  is  indeterminate  at  a  particular  point,  all 
the  lines  represented  by  the  complete  integral  pass  through  the 
point  in  question.  For,  if  the  point  be  a  node  of  the  particular 
integral  which  passes  through  it,  p  will  have,  at  this  point,  more 
than  one  value ;  and,  the  differential  equation  being  of  the  first 
degree,  this  can  only  happen  when  p  takes  the  indeterminate 
form.  For  example,  the  integral  of  xdy  +  ydx  =  o  is  xy  =  C, 
representing  a  system  of  hyperbolas  ;  but  the  particular  inte- 
gral which  passes  through  the  origin  is  the  pair  of  straight  lines 
xy  =z  o  o(  which  the  origin  is  a  node.  Accordingly  /  takes  the 
indeterminate  form  at  the  origin. 

'Ik 


Homogeneous  Equations, 

20.  The  differential  equation  of  the  first  order  and  degree, 
Mdx  -f  Ndy  =  o,  is  said  to  be  homogejieous  when  M  and  N  are 
homogeneous  functions  of  x  and  y  of  the  same  degree.  Since 
the  ratio  of  two  homogeneous  functions  of  the  same  degree  is  a 

function  oiZ,  a,  homogeneous  equation  may  be  written  in  the 

X 

form 

..        1  =  43 (■> 


§11.]  HOMOGENEOUS  EQUATIONS. 


y 

If,  now,  we  put  z/  for  -,  so  that 


dy  dv    , 

y  =  vx,     -^  =  X--  +  V, 
dx  dx 


X  "^^^  --      {^  "    ^ 


the  equation  becomes  '^ 

dv   ,  ,/  s 

dx 


a  differential  equation  between  x  and  v  in  which  the  variables 
can  be  separated  ;  thus, 

dx  dv 


X        f{v)  —  V 
21.  For  example,  the  equation 


{x^  —  y^)~  —  2xy  =  o (i) 

dx 


is  homogeneous.     Putting  ^  =  vx,  we  obtain 


dv    ,  2V 

X h  V 


dx  1  —  v^ 

whence 

dv  V  +  v^ 

X—  =  — ' , 

dx  I  —  v^ 

or 

dx          1  —  v^     y  dv         2vdv 

dv  =  — 


X        v{i  +  v^)  V        1  +  v^ 

Integrating, 


1 8  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  21, 


y 

and,  replacing  t^  by  -, 


or 


iog^i±J:!  =  c, 


x"  -\-  y^  =  cy (2) 

22.  The  geometrical  meaning  of  the  homogeneous  equation 

(i)  of  Art.  20  is  that  -/  has  the  same  value  for  all  points  at 
ax 

which  ^  has  a  given  value  \  that  is  to  say,  if  we  draw  a  straight 

X 

line  through  the  origin,  the  various  curves  of  the  system  repre- 
sented have  all  the  same  direction  at  their  points  of  intersection 
with  this  straight  line.  But  this  is  the  definition  of  similar  and 
similarly  situated  curves,  the  origin  being  the  centre  of  simili- 
tude. The  curves  of  such  a  system  are  simply  the  different 
curves  which  would  be  constructed  to  represent  the  same  equa- 
tion if  we  took  different  units  of  length.  Denoting  the  unit  of 
length  by  c,  the  general  equation  of  the  system  will  therefore 
be  of  the  form 


(rf)-»- 


/ 


where,  since  c  is 'arbitrary,  it  is  the  parameter  of  the  system  ; 
in  other  words,  the  constant  of  integration  c  may  be  so  taken 
that  the  complete  integral  of  the  homogeneous  differential 
equation  will  be  a  homogeneous  equation  between  x,  7,  and  c. 
In  the  example  given  in  the  preceding  article,  the  system  of 
curves  represented  consists  of  all  circles  touching  the  axis  of  x 
at  the  origin,  and  the  final  equation  is  so  written  that  all  of  its 
terms  are  of  the  second  degree  with  respect  to  x,  y,  and  c. 


§  IL]  HOMOGENEOUS  EQUATIONS.  1 9 


Case  in  which  the  Functions  M  and  N  are  of  the  First  Degree. 

23.  The  equation  Mdx  +  Ndy  =  o  can  always  be  solved  if 
M  and  N  are  functions  of  the  first  degree  in  x  and  y ;  that  is, 
when  the  equation  is  of  the  form 

dy  ^    ax  -\-  by  -\-  c  ^  , 

dx       a'x  +  by -h  c"     ' ^  ^ 


for,  substitute  in  this 


cC^-     cL    t. 


drj  _     a^  +  by]  -\-  ah  -\-  bk  +  c 


y  =  V  +  i,)  ^.^c{y^ 


and  we  have 

^^  x^d     L    ;i^    j_    ^7.    _l_    AA   _L    >- 

...        (3) 


d$       a'$  +  b'r]  +  a'h  -\-  b'k  ■\-  (f 
If,  now,  we  determine  h  and  k  by  the  equations 
ah   +  bk   -{-  c  =  o, 


•:::! « 


a'h  +  <^'/&  +  c' 


equation  (i)  takes  the  homogeneous  form 

dr]  a^  -\-  brj     , 

from  which  we  can  determine  the  integral  relation  between  i 
and  t] ;  and  thence,  by  substitution  from  (2),  the  relation 
between  x  and  y. 

24.  Equations  (4)  give  impossible  values  of  h  and  k  when  ^, 
b,  a',  and  ^'  are  proportional.     In  this  case,  putting 

a'  =  ma,     b'  =  7nb, 


20  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  24. 

equation  (i)  becomes 

dy  __      ax  -^  by  -\-  c 
dx       m{ax  +  by)  -f-  /* 

Now  put 

ax  -\-  by  —  z) 

whence 

dy  _  \  dz       a 

dx       b  dx       b' 

Making  these  substitutions,  we  have 

dz  ,    L  z  -^  c 

—  =z  a  -h  b 


dx  mz  -f-  ^' 

an  equation  in  which  the  variables  can  be  separated. 

Examples  II. 

Solve  the  following  differential  equations  :  — 

1.  (i  4-  x)ydx  +  (i  —  y)^dy  =  0,  \ogxy  =  c  —  x  -\-  y. 

"     "-^  T  '^. 

2.  -^  =  afXy  ax^y  +  O'  +  2  =  o. 
dx 

3.  {y^  +  xy^)dx  +  {x^  -  yx')dy  =  o,  log -^  =  c, 

4.  xy{i  H-  x^)dy  =  (i  +  y^)dx,  (i  +  ^^(i  -f-  jv^)  =  cx\ 

^jc  by  —  a 

dy       f  -\-  1  /      .        X 


§11.]  EXAMPLES.  21 

8.     sin  X  cos  y  .  dx  —  cos  x^vsxy  ,  dy,  cos  ^  =  ^  cos  x. 

Q.     ax^  +  2y  =  xy^,  x'^y''  =  cey. 

dx  dx 

,o.     ^  +  ^  +  ^^  +  r  ^  o,  ^+>^+^         ^  c 

dx       \  -\-  X  -\-  x^  2xy  +  X  +  y  —  1 

11.  ^  +  e^y  =  e^y^,  log-^  ~  ^  =  ^-^  +  ^. 

12.  ^^(I    -   /)  =   >'(!    +^),  log:>^_^~-^^  =   ^. 

13.  xdy  —  ydx  —  sjipc"  +  f)dx  =  0,  jc^  =  ^2  +  2cy, 

14.  (8);  H-  io.t)^^  +  (5;^  +  ^x)dy  ■=  o, 

{y  +  ^)^(.)'  +  2xy  =  ^. 

15-    (^  +  y)~-  +  ^  -  >'  =  o,       tan-^:^  +  41og(^2  +  ^2)  3^  ^, 
dx  X 

16.    (^>'-^0^  =  ;'%  '  ^=J. 


17.    ^  4.  ^^  =  2y,  \Qg{x  ^  y)  ^  c  - 

dx  X  —  y 

(;;  -  ;t  +  \y{y  +  ^  -  1)5  =  ^. 

19.  (:\:2  +  f)dx  —  2xydy  =  0,  jr^  —  j^  =  ex. 

20.  2^jj;^jt:  +  (_);2  —  TyX^^dy  =  0,  ^2  —  jj;^  =  ^jj;3. 

21.  y-"  +  (jcy  +  ^2)^  =  0,  ^>^  =  <^(^  -h  2j). 

22.  (^2  _  2,y^^xdx  +  (3-^  —  J>^)>'^  =  o> 


22  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE,   [Art.  2$. 


III. 
Exact  Differential  Equations, 

25.  An  exact  differential  containing  two  variables  is  an 
expression  which  may  arise  from  the  differentiation  of  a  func- 
tion of  X  and  y.     Denoting  the  function  by  Uy  we  have 

du='^'±dx  +  ^dy, '.     (I) 

ax  ay 

where  the  coefficients  of  dx  and  dy  are  the  partial  derivatives 
of  u.  Thus,  the  form  of  an  exact  differential  is  Mdx  +  Ndy. 
But,  if  M  and  N  are  any  given  functions  of  x  and  y^  we  cannot 

generally  put 

du  =  Mdx  +  Ndy) (2) 

for,  if  ^ 

M  =  '^f,    and    N  =  '^, (3) 

dx  dy 

we  must  have 

dM       dN  f  . 

-di^-d^^ (4) 

d^u 
because  each  member  of  this  equation  is  an  expression  for  — — - . 

dxdy 

Hence  equation  (4)  is  a  necessary  condition  of  the  possibility 

of  equation  (2)  or  equations  (3) ;   that  is,  of  the  existence  of  a 

function  whose  partial  derivatives  with  respect  to  x  and  y  are 

M  and  N  respectively. 

26.  It  is  also  a  sufficient  condition ;  for  the  most  general 
form  of  the  function  whose  derivative  with  respect  to  x  is  My  is 


'=( 


Mdx^Y, (5) 

where  Mdx  is  integrated  as  if  y  were  constant,  and  F  is  a 
quantity  independent  of  x,  but  which  may  be  a  function  of  y. 


§111.]  EXACT  DIFFERENTIAL   EQUATIONS.  23 

Now  the  only  other  condition  to  be  satisfied  is  that  the 
derivative  of  u  with  respect  to  y  shall  equal  N  \  that  is, 


dy]  dy 


or 


^  =  iV^  -  -^  f  Mdx (6) 

dy  dy] 

Since  Y  is  to  be  a  function  of  y  only,  but  is  otherwise 
unrestricted,  this  equation  merely  requires  that  the  second 
member  should  be  independent  of  x.  This  will  be  true  if  its 
derivative  with  respect  to  x  vanishes  ;  that  is  to  say,  if 

dN__dM_^^ 
dx'        dy 

This  equation  is  iflentical  with  equation  (4),  which  is,  therefore, 
a  sufficient,  as  well  as  a  necessary,  condition. 

27.  An  equation  in  which  an  exact  differential  is  equated  to 
zero  is  called  an  exact  differential  equation.  Using  the  notation 
of  the  preceding  articles,  the  complete  integral  of  the  equation 
Mdx  4-  Ndy  =z  o  when  exact  is  evidently 

u  =  C. 

The  function  u  is  determined  by  direct  integration  as  indicated 
in  equations  (5)  and  (6).  It  is  to  be  noticed  that  dV  consists 
of  those  terms  in  Ndy  which  do  not  involve  x ;  and  evidently 
the  integral  of  these  terms,  and  also  of  those  containing  x 
only,  may  be  considered  separately,  and  it  is  only  necessary  to 
ascertain  whether  the  terms  containing  both  x  and  y  form  an 
exact  differential.     For  example,  in  the  equation 

x(x  4-  2y)dx  +  (x""  —  y^)dy  =  o. 


24  EQUATIONS  OF  FIRST  ORDER  AND   DEGREE.   [Art.  2/. 

the  sum  of  these  terms  is  2xydx  -f  x'^dy^  which  is  the  differen- 
tial of  x^y ;  hence  the  complete  integral  is 

1^3  +  x-y  -  \y^  =  C, 
or 

x^  H-  z^y  —  y^  =  c. 

28.  An  expression  involving  only  some  function  of  x  and 
y,  and  the  differential  of  this  function,  is  obviously  an  exact 
differential.     Thus,  in  the  equation 

xdx  +  ydy        I    ydx  —  xdy  _ 
^{x'  -{-  r  -  i)        ~x^  +  y^     ~  °'  -^ 

the  first  term  is  a  function  of  x^  +  ^  and  its  differential,  and  the 
second  is  a  function  of  -  and  its  differential.  The  equation  may, 
in  fact,  be  written 


hence  the  integral  is 


<!)_„. 


2n/(^  +   _j;^   _    i)  /^y  ^   ^ 


y/(^  +  ^2_  i)  +  tan-'^  =  C. 


Integrating  Factors, 


29.  We  have  seen  in  the  preceding  articles  that  the  com- 
plete integral  of  an  exact  differential  equation  appears  in  the 

form 

u  =  C, (I) 

so  that  the  differential  equation  results  directly  from  the  differ- 
entiation of  the  integral,  C  disappearing  by  differentiation. 


§  III.]  INTEGRATING  FACTORS.  2$ 

Now,  since  the  integral  of  any  equation  can  be  put  in  the 
form  (i)  by  solving  it  for  C,  it  follows  that,  whenever  we  can 
solve  an  equation  of  the  form 

Mt/x  +  N(/y  =  o, (2) 

we  can  produce  an  exact  differential  equation  which  is  equiva- 
lent to  the  given  equation,  that  is  to  say,  which  is  satisfied  by 
the  same  simultaneous  values  of  x,  y,  and  /,  This  new  differ- 
ential equation  being  of  the  first  order  and  degree,  must  then 
be  of  the  form 

y.{Mdx  -t-  Ndy)  =  0, (3) 

where  /a  is  a  factor  containing  ;ir  or  _;^  or  both,  but  not  containing 

/. 

The  factor  /x,  which  converts  a  given  differential  equation 
into  an  exact  differential  equation,  is  called  an  ijitegrating 
factor. 

For  example, 'solving  equation  (2)  of  Art.  21  for  ^,  we  have 

c^y  +  -y 
whence,  differentiating, 

2xydx  —  x^dy       2xydx  +  {y^  —  x^)dy 
o  =  dy  + = > 

and,  comparing  this  with  equation  (i)  of  Art.  ^i,  we  see  that 
—  is  an  integrating  factor  of  that  equation. 

30.  A  differential  equation  has  a  variety  of  integrating 
factors  corresponding  to  different  forms  of  the  complete  inte- 
gral.     For  example,   one   integrating   factor   of   equation   (i), 


26  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  30. 

Art.  17,  is  the  factor  by  means  of  which  we  separated  the 
variables ;  namely, 

I , 

(I  +a:)(i  -  yY 

and  this  corresponds  to  the  form  (3)  of  the  integral ;  but,  if  we 
differentiate  the  integral  in  the  form  (5),  Art.  18,  wgl  obtain 
equation  (i)  multiplied  by  the  integrating  factor 


(I  -  yy 

The  forms  of  the  integral  differ  in  respect  to  the  constants 
which  they  contain.  In  general,  \i  ti  ^=-  c  is  an  integral  giving 
the  integrating  factor  /n,  so  that 

du  —  ^{Mdx  +  Ndy)y 
then 

f{u)  =  C 

where  C  =  f{c)  is  another  form  of  the  integral ;  and  this  gives 
the  exact  differential  equation 

f{u)du  =  o, 
or 

f{u)iL{Mdx  +  Ndy)  =  o. 

Hence  f{ti)\x.  is  also  an  integrating  factor ;  and,  since  /"denotes 
an  arbitrary  function,  /'  is  also  arbitrary ;  thus,  the  number  of 
integrating  factors  is  unlimited. 

31.  The  form  of  the  given  differential  equation  sometimes 
suggests  an  integrating  factor.     For  example,  in  the  equation 

(j  +  \ogx)dx  —  xdy  —  o, 


§111.]  INTEGRATING  FACTORS.  2/ 

the  terms  containing  both  x  and  y  are 

ydx  —  xdy. 

This  expression  becomes  an  exact  differential  when  divided 
either  by  y^  or  by  x^.     The  remaining  term  contains  x  only ; 

hence  —  is  an  integrating  factor.     Thus,  we  write 
ydx  —  xdy       Xogxdx 

whence,  integrating, 

—y       log  X       I 

^  X  X  ^     ' 

or 

<:^  +  J/  -h  log:v  H-  I  =  o. 

32.  The  expression  j/<3r;f  —  xdy,  which  occurs  in  the  preced- 
ing article,  is  a  special  case  of  a  more  general  one  which  should 
be  noticed.     For,  since 

d{x^y^)  =  x^-'^y^-'^{mydx  +  nxdy), 

an  expression  of  the  form 

x^y^{mydx  +  nxdy) (i) 

has  the  integrating  factor 

jQtn  —  i—ayn  —  i—P  • 

and  since,  by  Art.  30,  the  product  of  this  by  any  function  of  //, 
where  u  =  x'y*",  is  also  an  integrating  factor,  we  have  the  more 
general  expression 

^km  —  x—a-ykn—r—^ ^2) 

(in  which  k  may  have  any  value)  for  an  integrating  factor. 


28  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  32. 

As  an  illustration,  take  the  equation 

yi^y^  4-  2x^^dx  -f  x{jx^  —  2y^)dy  5=  o. 
This  may  be  written  in  the  form 

y^(ydx  —  2X£fy)  +  x*(2ytfx  +  xdy)  =  o, 
in  which  each  term  is  of  the  form  (i).     In  the  first  term, 

m  =  I,    «  =  —  2,    a  =  o,    ^  =  3; 
and  the  expression  (2)  gives,  for  the  integrating  factor, 

that  is  to  say,  any  multiplier  of  this  form  will  convert  the  first 
term  into  an  exact  differential.  In  like  manner,  any  expression 
of  the  form 

X2k'-5yk'-i 

is  an  integrating  factor  of  the  second  term.  A  quantity  which 
is  at  once  of  each  of  these  forms  will  therefore  be  an  integrating 
factor  of  the  given  equation.  Equating  the  exponents  of  x^ 
and  also  the  exponents  of  j,  in  the  two  expressions,  we  have 

—  2^  —  4  =    k'  —  \j 

from  which  k  ^  —2,  and  the  integrating  factor  is  x-^. 
Multiplying  the  given  equation  by  ;t:-3,  we  have 

y^x-'^dx  —  2y^x-^dy  +  2xydx  +  x^dy  =  o; 


§  III.]  EXAMPLES.  29 

and,  integrating, 

or 

20c^y  —  y^  —  cx^. 

Examples  III. 
Solve  the  following  differential  equations  :  — 

I.    {pc^  —  ^xy  —  2y^)dx  -\-  i^f  —  a^xy  —  2x'^)dy  =  o, 

xi  -^  yi  —  dxyix  -\-  y^  z=z  c. 

>    2.    --^  =  ^ ~  ^  ,  xy"^  =  x^y  4-  ^. 

rt!^        X  3jv^  —  JC 

.  3.     (2:^  —  J  +  i)^:'c  +  {2y  —  X  —  i)dy  =  o, 

oc^  —  xy  •\-  y^  •\-  X  —  y  =^  c. 

,  4.     ^(:r2  4-  2>y^)dx  +  y{y^  +  3^)^  =  o, 

X*  +  6:r2>'2  -}.  y^  =  c. 

xdy  —   r^;c                      ^  +   v^  y 

,  5.     ^//j;  +  xdx  +     ^a  ^  ^2     =  o, —  +  tan-'-  =  c. 

V  6.    (y  —  x)dy  +  ydx  =  o,  logy  -h  -  =  c, 

V  7.     ax-^y""-^  =  2^-=^  —  y,  ^ =  -^  +  ^. 

'  -^  ^:x:  dx       ^  n  -h  2        X 

.  8.     :v^  —  y  =  ^(x^  —  y^),  sin-'=^  =  logx  -he, 

uX  X 

9.     x^-  y  =  xs/{x-  +  y),  y  =  ^(^^-  -  -^V 

dx  2\  ce^  I 


30  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.    [Art.  32. 

v/   10,     Q.<ys,ax-^  -I-  ay''ivs\ax  —  x, 
ax 

a^y  =  ax  sin  ax  +  cos  ax  .  log  cos  ax  -\-  c  cos  ax, 

'    II.  (^3  _  2yx^)tfx  4-  (2jc>^  —  AT^)^  =  o,          x'y^  =  jv^^c'*  +  ^. 

.12.  (2jc2jj''  +  y)t/x  —  {x^y  —  sx)dy  =  o,      /[x^y  =  5  +  i:x^y^. 

^3-  (/*  —  2jc3j)rt'jc  +  (;d  —  2JC>'3)^  =  0,             ^3  -j-  ^3  =  ^;cy. 

14.  Solve  Exs.  II.,  19  and  20,  by  means  of  integrating  factors. 


IV. 

T/ie  Linear  Equation  of  the  First  Order. 

33.  A  differential  equation  is  said  to  be  linear  when  it  is  of 
the  first  degree  with  respect  to  y  and  its  derivatives.  The 
linear  equation  of  the  first  order  may  therefore  be  written  in 
the  form 

ax 

where  P  and  Q  denote  functions  of  x. 

Consider  first  the  case  in  which  the  second  member  is  zero, 
that  is  to  say,  the  form 

^  +  ^  =  0 (I) 

ax 

The  variables  may  be  separated  ;  thus, 
^  =  ^Mx, 

y 


§  IV.]        LINEAR  EQUATION  OF  THE  FIRST  ORDER.  3 1 


Hence 
or 


log  J  =  ^  —  \PdXf 

y=^Ce-\'" .     (2) 

In  this  solution,    Pdx  may  be  taken  to  denote  a  particular 

integral  of  Pdx^  since  the  constant  is  directly  expressed  in  the 
equation. 

34.  If  we  put  equation  (2)  in  the  form 

and  differentiate,  we  have 

eV'^^'^dy  +  Pydx)  =  o, 

which  shows  that  e^^^  is  an  integrating  factor  of  equation  (i). 
Since  Q  is  a.  function  of  x  only,  it  follows  that  e^  ^'^^  is  also  an 
integrating  factor  of  the  more  general  equation 

-f  +  iy  =  Q (') 

ax 
Hence,  to  solve  this  equation,  we  write 


e 
and,  integrating, 

e 


^^^y^^U^'^^Qdx  +  C, (2) 


In   a  given   example,  the   integral?   involved   in   the   general 
expression    (2)    should,    of    course,    be   evaluated    if    possible. 


32  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  34. 

Thus,  let  the  given  equation  be 


(I 

+-)£= 

w  +  xy, 

^. 

*           V 

in 

dx 

I  +  ^'■' 

I  +^ 

X 

tVif  rpfrvrf* 

or 


Here  P  = 

jPdx=  -ilog(i  H-^0» 
and 

J^'^-  = I 

is  the  integrating  factor.     Hence 

I  X  mdx 


and 


*'*        +  C, 


or 


J  =  WJt:  +  Csl{i  +  ^). 

Transformation  of  a  Differential  Equation. 

35.  It  is  frequently  useful  to  transform  a  given  differential 
equation  by  replacing  one  of  the  variables  by  a  new  variable 
which  is  an  assumed  function  of  the  variable  replaced,  or  of 
both  variables.     The  form  of  the  assumed  function  is  generally 


§  iV.]  EXTENSION  OF   THE  LINEAR  EQUATION.  33 

suggested  by  that  of  the  given  equation.     Thus,  the  form  of 
the  equation 

(i  -f  xy)ydx  +  (i  —  xy)xdy  =  o 

suggests  the  use  of  a  new  variable  v  =  xy^  whence 

xdy  =.  dv  —  ydx. 
Eliminating  y,  we  have 

(i  +  v)-dx  +  (i  —  v)dv  —  (i  —  v)-dx  =  o, 

•  X  X 

or 

2Tfdx  +  (i  —  v)xdv  =  o, 

in  which  the  variables  v  and  x  can  be  separated.     Integrating, 


2  log  .;c  —  -  —  log  z;  =  ^, 
or 

and,  substituting  xj^  for  ^, 

x  =  Cye^y, 

Extension  of  the  Linear  Equation. 
36.  If  z;  =  f{y)i  the  linear  equation  for  Vy 


becomes 


dx 

f{y)f^  +  Ff{y)  =  C (0 


In  other  words,  an  equation  of  this  form  becomes  linear  if  we 
put  V  =  f{y). 


34  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE.   [Art.  3/. 

For  example,  the  equation 

^+MX_(^_,)sec;-  =  o 
dx       ^  +  I 

takes  the  form  (i)  when  multiplied  by  cos^,  and  hence  is  a 
linear  equation  for  sin  y.     The  integral  will  be  found  to  be 

sin;;  =  ^^  -  3^  +  ^. 
3(^  +  I)      . 

37.  In  particular,  the  equation  r^Ji/^ 

^-^jy  =  Qr (2) 

is  known  as  the  extension  of  the  linear  equation.     Dividing  by 
y**,  we  have 

ax 
or 

(I  _  n)y-n^  +  (I  -  n)Py^—  =  (i  -  n)Q, 
ax 

which  is  of  the  form  (i),  and  therefore  linear  for  y^-^. 


Examples  IV. 

Solve  the  following  linear  equations  :  — 

dv 
^  \.     -^  •\-  y  =■  Xf  v  =  a:—  1+  ce-^, 

dx 

2.    -f  =  by  +  asmxy  y  =  ce^''  —  a '-^ . 

dx  1  -{-  o^ 

,3.    ^ ^J—  =  {x  +  1)3,  2y  =^{x  +  lY  +  c{x  +  ly. 

ax       X  +  1 


§  ly.]  EXAMPLES.  35 

4.    &  ^  n^  =:  ^^jf«,  y  =  a;«(<?^  +  c). 

dx  X 


ax 


6.    -^cos^  +  7sin^  =  I,  y  =  smx  +  ccosx. 

dx 


>/  7.     -^  +  y cos ^  =  sin 2;*:,  y  =  2smx  —  2  +  ce- 

dx 


smjr 


8.      ^-^  —  ^V  =  :x:  +  I,  y  =  — +  cx^. 

dx  1  —  a       a 


Y    9-      -T^  + 


i^  4.    ^y    = 


^jx:    I  +  ^    2^(1  +  x^y 

_  log  [V^(i  +  ^')  -  i]  -  log^  +  ^ 

•^  ~"  2sJ{l    +  ^^2) 


10.     ;v(l  —  ^2)_J:  _|>  (2a:2  —  i)jv  =  ^:r3, 


jj;  =  ^^  -f  cx\]{i  —  x^). 


II.    cosx-^  +  V  —  I  +  sinjv  =  o,     y{sQCx  +  i3Xix)  =  x  +  c 
dx 


13'    (i  +  >'0^^  =  (tan-*^  —  x)dy, 

.      ;«:  =  tan-^>'  —   I   +  r^-tan-'j- 


36  EQUATIONS  OF  FIRST  ORDER  AND  DEGREE,   [Art.  37. 

Solve,  by  transformation,  the  following  equations  :  — 


14.     -^  =  w^  +  «V  +  /, 
dx 


mnx  +  ify  ■\-  m  •}-  pn  •=  ce"^. 


y2 

Y     15.    {x  —  y)dx  +  2xy£fy  =  0,  logx  +  -^  =  c. 


16.    (x-h  yy^  =  a', 

ax 


=  tan 


17.    ^  +  >'  =  ^>'^  ;^  =  ^  +  i  +  ^^'" 


18.        -^   =   ^31,3    ^   xy,  -L    =:   ^2  _^    I    +    ^^-^'. 

fl'jc  y^ 

19/(1  -  A^)-^  -  ^j;  =  axf,  ~  =  rv/(i  ~  x^)  -  a, 

dx  y 


20.      3/^  _  ^^3  =  ^  +   I,  yl  =  ^^^  -  ^^±JL  _  -i 


dx  a  a 


21.     x^  +  JJ'  =  /logjt:,  -  =  log^  •\-  \  -\-  ex, 

dx  y 

dx       2(1  —  ^2)        y 

23.       ^  =  1 ,  i  =    2    _  y  +  ,,-*/. 

'     dx      xy  +  X'y^  x 


§  v.]  DECOMPOSABLE  EQUATIONS.  37 


CHAPTER    III. 

EQUATIONS  OF  THE  FIRST  ORDER,  BUT  NOT  OF  THE  FIRST  DEGREE. 

V. 

Decomposable  Equations. 

38.  A  differential  equation  of  the  first  order  is  a  relation 
between  x,  7,  and  /.  If  the  equation  is  not  of  the  first  degree 
with  respect  to  /,  the  first  step  in  the  solution  is  usually  to 
solve  the  equation  for  /.  Suppose,  in  the  first  place,  that  the 
equation  is  a  quadratic  in  / ;  then  two  values  of  /  in  terms  of 
X  and  y  are  found.  These  will  generally  be  irrational  functions 
of  X  and  y ;  in  the  exceptional  case  when  they  are  rational 
functions,  the  equation  will  be  decomposable  into  two  equations 
of  the  first  degree.     For  example,  the  equation 

may  be  written 

(l-X2-')=°- 


and  is  satisfied  by  putting 


dx 


X  —  o (2) 


or 

^-y  =  <^ (3) 

ax 


38  Equations  not  of  the  first  degree.  [Art.  38. 

The  integrals  of  these  equations  are 

2y  =  x^  -\-  c (4) 

and 

y  =  Ce- (5) 

respectively.     Each  of  these  is  therefore  an  integral  of  equa- 
tion (i). 

Thus,  a  decomposable  equation  of  the  second  degree  has  two 
distinct  solutions. 

Equations  Properly  of  the  Second  Degree. 

39.  In  a  proper,  that  is,  an  indecomposable,  equation  of  the 
/Second  degree,  the  two  expressions  for  /  are  the  values  of  a 
two-valued  function  of  x  and  y  expressed  by  attaching  the 
ambiguous  sign  to  the  radical  involved.  There  is,  in  this  case, 
6ut  one  integral,  the  ambiguity  disappearing  in  the  process  of 
mtegrating  or  of  rationalizing  the  result ;  so  that  it  is,  in  fact, 
unnecessary  to  retain  the  ambiguous  sign  in  the  expression  for 
/.     For  example,  the  equation 


gives  /  =   ±^;  whence 


fdyV  ^  y 
\dxj         X 


(0 


dx    ,    dy 
sjx       sjy 
and,  integrating, 

^x  ±  ^y  =   ±\Jc (2) 

> 

But,  rationalizing  this,  we  have 

{x  —  yy  -  2c{x  +  y)  +  c-  —  o,    .     .     .     .     (3) 

a  single  equation  for  the  complete  integral. 


§  v.]  INDECOMPOSABLE   EQUATIONS.  39 

The  system  of  curves  represented  by  equation  (3)  consists 
of  parabolas,  each  of  which  touches  the  two'  axes  at  the  same 
distance  c  from  the  origin,  and  the  different  combinations  of 
signs  in  equation  (2)  simply  correspond  to  the  three  arcs  into 
which  the  parabola  is  separated  by  the  points  of  contact. 


Systems  of  Curves  corresponding  to  Equations  of  Different  Degrees, 

40.  A  differential  equation  of  the  first  degree  is,  properly 
speaking,  one  in  which  /  has  a  single  value  for  given  simulta- 
neous values  of  x  and  y.  .  An  equation  of  the  second  degree 
is  one  in  which  /  has  two  values  for  given  values  of  x  and  j, 
and  so  on.  Thus,  such  an  equation  as  /  =  sin-'ji:  is  not  an 
equation  of  the  first  degree,  because,  for  any  value  of  x,  sin-';r 
has  an  unlimited  number  of  values.  The  general  form  of  an 
equation  of  the  first  degree  is,  then, 

Z/  +  J/  =  o, 

in  which  L  and  M  denote  one-valued  functions  of  x  and  y. 

Two  curves  of  the  system  cannot,  in  general,  intersect,  for 
in  that  case  there  would  be  two  values  of  /  at  the  point  of 
intersection.  The  points,  if  there  be  any,  at  which  L  =:  o  and 
M  =  o,  form  an  exception ;  for,  at  these  points,  /  is  inde- 
terminate, as  exemplified  in  Art.  19.  Thus,  the  curves  of  the 
system  either  do  not  intersect  at  all,  or  intersect  only  at  certain 
points  where  /  is  indeterminate.*  It  follows  that,  in  the  integral 
equation,  given  simultaneous  values  of  x  and  y  must,  except  in 

*  The  same  reasoning  shows  that,  the  differential  equation  being  of  the  first 
degree,  points  in  which  two  arcs  corresponding  to  the  same  value  of  c  intersect,  in 
other  words,  nodes,  can  only  occur  at  the  points  where  /  is  indeterminate.  Con- 
versely, these  points  must  either  be  points  through  which  all  the  curves  of  the 
system  pass,  or  else  nodes.  In  the  latter  case,  they  may  be  crunodes  through  which 
two  real  arcs  of  a  particular  integral  pass,  or  acnodes  through  which  no  arc  passes. 


40  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.   [Art.  40. 

the  case  of  the  points  above  mentioned,  determine  a  single  value 
of  c,  or,  at  least,  values  of  c  which  determine  a  single  curve. 
For  example,  the  integral  of  the  equation 

/=  I  +y (i) 

is 

Xzxi-^  y  —  X  —  o. (2) 

If  we  give  particular  values  to  x  and  7,  we  find  an  unlimited 
number  of  values  of  a  differing  by  multiples  of  tt  ;  but,  writing 
the  equation  in  the  form 

y  =  tan(jf  4-  a), 

we  see  that  these  values  determine  but  a  single  curve.  We,  in 
fact,  obtain  all  the  curves  of  the  system  by  allowing  a  to  range 
in  value  from  o  to  tt  ;  and,  as  a  varies  over  this  range,  the  curve 
sweeps  over  the  whole  plane  once. 

If  we  take  the  tangent  of  each  member  of  equation  (2),  and 
write  tan  a  =  ^,  we  have 

y  —  tan^   _ 
I  +  _ytan;t' 

in  which  any  simultaneous  values  of  x  and  y  determine  a  single 
value  of  c ;  and  c  must  pass  over  the  range  of  all  real  values  in 
order  to  make  the  curve  sweep  once  over  the  entire  plane. 

41.  In  general,  if  the  constant  of  integration  is  such  that 
different  values  of  it  always  correspond  to  different  curves, 
there  can  be  but  one  value  of  c  for  each  point ;  hence  the  form 
of  the  integral  is 

/{:  4-  (2  =  o 

V  where  P  and  Q  are  one-valued  functions  of  x  and  y^  and  this 
we  may  regard  as  the  standard  form  of  the  integral.  It  will  be 
noticed  that  both  /*  =  o  and  Q  =  o  sltq  particular  integrals  ; 


§  v.]  SYSTEMS  OF  INTERSECTING    CURVES.  4I 

the  former  corresponding  to  ^  =  00,  and  the  latter  to  c  =z  o. 
Thus,  in  the  example  given  above,  j/  =  tan  x  and  jy  =  —cot  x 
are  particular  integrals. 

42.  In  like  manner,  the  form  of  the  differential  equation  of 
the  first  order  and  second  degree  is 

Lp'  -^  Mp  -{-  N  =  o, 

where  L,  M,  and  N  are  one-valued  functions  of  x  and  j/.  In 
general,  two  curves,  and  two  only,  representing  particular 
integrals,  intersect  in  a  given  point.  When  the  expression 
Z/2  _(-  Mp  +  N  can  be  separated  into  rational  factors  of  the 
first  degree,  these  curves  belong  to  distinct  systems  having  no 
connection  with  one  another,  as  in  Art.  38 ;  but,  in  the  general 
case,  they  are  curves  of  the  same  system.  Thus,  the  system  of 
curves  representing  a  proper  equation  of  the  second  degree  is 
a  system  of  intersecting  curves,  two  curves  of  the  system,  in 
general,  passing  through  a  given  point.  Hence,  in  the  integral 
equation,  given  simultaneous  values  of  x  and  j/  must  generally 
determine  two  values  of  c,  or,  at  least,  values  of  c  which  deter- 
mine two  and  only  two  curves  of  the  system. 

43.  Take,  for  example,  the  equation 

/>'  =  ^  -  y^ (i) 

or 

—-^ ^dx. 

±v/(i  -  r) 

The  integral  is 

sin-'y  —  X  =  a, .     (2) 

or 

y  =  sin(x  -\-  a), (3) 

in  which  it  is  permissible  to  drop  the  ambiguous  sign,  because 
j^  =  —  sin  (^  -I-  a)  may  be  written  j/  =  sin  (x  -^  -n-  -j-  a),  and  is 
therefore  included  in  the  integral  (3).  Here,  as  in  the  example 
of  Art.  40,  if  we  give  particular  values  to  x  and  j/,  a  has  an 


42  EQUATIONS  NOT  OF  THE  FIRST  DEGREE.   [Art.  43. 

unlimited  number  of  values ;  for,  if  6  be  the  primary  value  of 
sin-' J,  every  expression  included  in  either  of  the  forms 

2«7r  +  ^     or     (2;z  +  i)7r  —  ^, 

where  n  is  an  integer,  is  a  value  of  sin  -  ^x.  These  values  of  a, 
however,  determine  but  two  distinct  curves,  since  values  of 
a  differing  by  a  multiple  of  2tt  determine,  in  (3),  the  same  curve, 
so  that  each  of  the  above  forms  determines  but  one  curve. 
Equation  (3),  in  fact,  represents  the  system  formed  by  moving 
the  curve  of  sines,  y  =.  sin  ;r,  in  the  direction  of  the  axis  of  x, 
and  we  obtain  all  the  curves  of  the  system  while  a  varies  from 
o  to  27r,  each  branch  or  wave  of  the  curve  falling,  when  a  =  27r, 
upon  the  original  position  of  an  adjacent  branch.  In  this 
motion,  the  curve  sweeps  twice  over  that  portion  of  the  plane 
which  lies  between  the  straight  lines  y  =  i  and  j/  =  —i,  for 
which  portion  only  the  value  of  /  is  possible  in  equation  (i). 

44.  If,  in  the  integral  of  an  equation  of  the  second  degree,, 
we  so  take  the  constant  of  integration  c,  that  different  values 
of  it  always  correspond  to  different  curves  of  the  system,  there 
can  be  but  two  values  of  c  corresponding  to  a  given  point.  The 
equation  will  then  take  the  form 

Pc^  +Qc  +  R  =  o 

where  P,  Q,  and  R  are  one-valued  functions  of  x  and  y ;  and 
this  may  be  regarded  as  the  standard  form  of  the  integral. 

To  reduce  equation  (3)  of  the  preceding  article  to  the  stand- 
ard form,  we  have,  on  expanding, 

y  =  sin  ^  cos  a  -}-  cos  :r  sin  a, 

in  which  sin  a  and  cos  a  are  to  be  expressed  in  terms  of  a  single 
constant.  For  this  purpose,  we  do  not  put  sin  a  =  r  and 
cos  a  =  v^(i  —  c^),  because  this  would  require  us  to  introduce 


§  v.]  SINGULAR  SOLUTIONS.  43 

an  irrelevant  factor  in  rationalizing  the  equation  in  c ;  but  we 
express  sin  a  and  cos  a  by  the  rational  expressions  the  sum  of 
whose  squares 'is  unity;  that  is,  we  put 

\    —   C^  2C 

sin  a  =  ,  cos  a  = 


I    +   (T^  I    +   r^ 

We  thus  obtain 

(c^^y  +  cos  x)  —  2r  sin  x  +  y  —  cos  x  =  o, 

which  is  the  complete  integral  of  equation   (i),  Art.  43,   ex- 
pressed in  the  standard  form. 


Singular  Solutions. 

45.  Representing  a  set  of  simultaneous  values  of  x,  y,  and 
/  by  a  moving  point,  every  moving  point  which  satisfies  a 
given  differential  equation  is,  at  each  instant,  moving  in  some 
one  of  the  systems  of  curves  representing  the  complete  integral. 
In  this  sense,  the  latter  completely  corresponds  to  the  differen- 
tial equation  :  nevertheless,  there  are,  in  some  cases,  other 
curves  in  which,  if  a  point  be  moving,  it  will  satisfy  the  differ- 
ential equation.  For,  suppose  a  curve  to  exist  which,  at  each 
of  its  points,  touches  one  of  the  curves  representing  particular 
integrals ;  then  a  point  moving  in  this  curve  is  moving  at  each 
instant  in  the  same  direction,  that  is,  with  the  same  value  of  /, 
as  if  it  were  moving  in  a  curve  representing  a  particular  inte- 
gral ;  hence  it  satisfies  the  differential  equation. 

Such  a  curve  is  an  envelope  of  the  system  of  curves  repre- 
senting the  complete  integral,  and  its  equation  is  called  a 
singular  solntion  of  the  differential  equation.  An  example  has 
already  been  noticed  in  Art.  8  ;  the  equation 

xf'  ^  yp  -\-  a  =.  o 


44  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.   [Art.  45. 

has  the  singular  solution 

in  addition  to  the  complete  integral 

y  ^  ex  -ir-. 
c 

Now,  the  latter  is  the  general  equation  of  the  tangents  to  the 
parabola  y^  =  4ax,  which  accordingly  form  a  system  of  straight 
lines  of  which  the  singular  solution  represents  the  envelope. 

46.  We  shall  now  show  how  a  singular  solution,  when  it 
exists,  may  be  found,  either  directly  from  the  differential  equa- 
tion, or  from  the  complete  integral  if  the  latter  be  known. 
Two  curves  of  the  system  touching  the  envelope  at  neighboring 
points  intersect  in  a  point  which  ultimately  falls  upon  the 
envelope  when  the  two  curves  are  brought  into  coincidence ; 
hence  the  envelope  is  sometimes  called  the  /ocus  of  the  inter- 
section of  consecutive  curves.  While  the  curves  are  distinct, 
two  values  of  /  in  the  differential  equation  correspond  to  the 
point  of  intersection.  These  become  equal  when  the  curves 
coincide,  that  is,  when  the  point  is  moved  up  to  the  envelope  ; 
and  they  become  imaginary  when  the  point  crosses  the  envel- 
ope. In.  like  manner,  if  we  substitute  the  coordinates  of  the 
point  in  the  integral  equation,  it  determines  two  values  of  c 
while  the  curves  are  distinct ;  and  these  become  equal  when 
the  point  is  moved  up  to  the  envelope,  and  imaginary  when  the 
point  crosses  it.  Thus,  at  every  point  of  the  envelope,  both 
the  differential  equation  considered  as  an  equation  for  /,  and 
the  integral  equation  considered  as  an  equation  for  Cy  have  a 
pair  of  equal  roots.  Hence,  if  we  form  the  condition  for  equal 
roots  in  either  of  these  equations,  which  we  shall,  for  shortness, 
call  the  /-equation  and  the  r-equation,  we  shall  have  an  equation 
which  must  be  satisfied  by  every  point  on  the  envelope. 


§  v.]  THE  DISCRIMINANT.  45 

47.  The  expression  which  vanishes  whenever  two  roots  of 
an  equation  are  equal  is  called  the  discriniijiant  of  the  equation. 
The  discriminants  of  the  /-equation  and  the  ^-equation  are 
expressions  involving  x  and  y.  Either  of  these  expressions 
may  break  up  into  factors,  the  vanishing  of  any  one  of  which 
causes  the  discriminant  to  vanish.  Hence  it  follows  from  the 
preceding  article  that,  if  there  be  a  singular  solution,  its  equa- 
tion is  the  result  of  putting  the  /-discriminant,  or  one  of  its 
factors,  equal  to  zero,  and  it  is  likewise  the  result  of  putting  the 
^-discriminant,  or  one  of  its  factors,  equal  to  zero. 

For  example,  in  equation  (i),  Art.  43,  the  condition  for  equal 
roots  is  evidently  y'^  —  i  —  o  ;  hence 

7—1  =  0    and     y  -\-  \  —  o 

are  the  only  equations  which  can  possibly  be  singular  solutions. 
Each  of  these  equations  gives,  by  differentiation,  /  :=  o,  and 
is  found  to  satisfy  the  differential  equation.  Hence  they  are 
singular  solutions,  the  lines  they  represent  being  envelopes  of 
the  sinusoids  represented  by  the  complete  integral. 

48.  The  general  method  of  finding  the  discriminant  of  an 
equation  is  to  differentiate  it  with  respect  to  the  unknown 
quantity  and  then  to  eliminate  that  quantity  between  the 
result  and  the  original  equation.  But,  in  the  case  of  the  equa- 
tion of  the  second  degree,  it  is  found  more  simply  by  solving 
the  equation.     Thus  the  /-equation,  in  this  case,  is 

Lf  +  Mp  +  N  =  o; (i) 

whence 

f=^K.±JiMlszAMl. (2) 

2Z 

so  that  the  condition  for  equal  roots  is 

M"^  -  ^LN  =  o .     (3) 


46  EQUATIONS  NOT  OF   THE  FIRST  DEGRJEE.      [Art.  48. 

In  like  manner,  the  general  form  of  the  c-equation  is 

and  the  condition  for  equal  roots  is 

Q"  -  A,PR  ==  o. 

For  example,  in  the  final  equation  of  Art.  44,  the  condition  for 
equal  roots  is 

4  sin^  X  —  4  ( j^  —  cos^  x)  =  o, 
or 


which  is  identical  with  the  like  condition  for  the  /-equation 
given  in  Art.  47. 

Cusp-Loci. 

49.  There  are  other  loci,  for  points  upon  which  the  discrimi- 
nants vanish,  which  it  is  necessary  to  distinguish  from  the 
envelope  whose  equation  alone  is  a  singular  solution.  There 
is,  in  fact,  no  reason  why  the  values  of  p  derived  from  the 
differential  equation,  when  they  become  equal  as  the  point 
(jT,  y)  crosses  a  certain  locus,  should  also  become  equal  to  the 
value  of  /  for  a  point  moving  along  that  locus.  Suppose,  then, 
that  the  two  arcs'  of  particular  integral  curves  passing  through 
U',  y)  meet,  without  touching,  the  locus  for  which  the  values 
of  p  become  equal ;  and  suppose,  as  will  usually  be  the  case, 
that  the  values  of  /  become  imaginary  as  we  cross  the  locus  ; 
then,  when  {x^  y)  is  moved  up  to  the  locus,  the  two  arcs  will 
come  to  have  a  common  tangent ;  and,  since  they  cannot  cross 
the  locus,  they  will  form  a  cusp,  becoming  branches  of  the 
same  particular  integral  curve.    Thus,  the  two  values  of  c  which 


§  v.]  CUSP-LOCI.  47 

corresponded  to  the  two  intersecting  arcs  will   also  become 
identical,  and  the  locus,  which  is  called  a  cusp-locus,  is  one  for 
which  the  ^-discriminant  also  vanishes. 
For  example,  the  roots  of  the  equation 

are  equal,  each  being  equal  to  zero,  when 

^  =  o; 

but,  since  p  =z  co  for  a  point  moving  along  this  line,  this  equa- 
tion does  not  satisfy  the  differential  equation.  The  complete 
integral  is 

{y  +  cy  =  x\ 

in  which  the  condition  of  equal  roots  is  x^  =■  o.  The  system 
of  curves  is  that  resulting  from  moving  the  semi-cubical  par- 
abola j/^  =  x^,  which  has  a  cusp  at  the  origin,  in  the  direction 
of  the  axis  of  y.     This  axis  is,  therefore,  a  cusp-locus. 

Tac-Loci  and  Node-Loci. 

50.  In  the  preceding  article,  the  values  of  /  were  supposed 
to  become  imaginary  as  we  cross  the  locus  for  which  they 
become  equal.  From  equation  (2)  of  Art.  48,  it  appears  that 
this  will  be  the  case  if  the  discriminant  changes  sign,*  but 
otherwise  not ;  hence,  if  the  factor  which  vanishes  at  the  locus 
appears  in  the  /-discriminant  with  an  even  exponent,  /  will  not 
become  imaginary  in  crossing  the  locus.  In  this  case,  the  two 
intersecting  arcs  cross  the  locus  ;  and,  when  {x,  y)  is  moved  up 
to  the  locus,  we  shall  simply  have  two  particular  integrals  which 
touch  one  another.     Such  a  locus  is  called  a  tac-lociis.     Since 


*  Since  the  discriminant  is  the  product  of  the  squares  of  the  differences  of  the 
roots,  this  will  be  true  also  for  equations  of  the  third  and  higher  degrees. 


48  EQUATIONS  NOT  OF   THE   FIRST  DECREE.      [Art.  50. 

the  values  of  c  for  the  two  curves  remain  distinct,  the  factor 
indicating  a  tac-locus  does  not  appear  at  all  in  the  r-discrimi- 
nant,  but  appears  in  the  /-discriminant  with  an  even  exponent.* 

51.  On  the  other  hand,  a  factor  may  appear  in  the  ^-discrimi- 
nant with  an  even  exponent,  and  not  at  all  in  the  /-discriminant. 
Through  every  point  of  the  locus  on  which  such  a  factor  van- 
ishes, the  proper  number  of  arcs  of  particular  integral  curves 
pass,  but  two  of  them  correspond  to  the  same  value  of  c ;  thus, 
the  point  is  a  node  of  the  curve  determined  by  this  value  of  Cy 
and  the  locus  is  called  a  node-loctis, 

52.  The  equation 

xp^  —  {x  —  ay  —  o (i) 

furnishes  an  example  of  each  of  the  cases  mentioned  in  the  two 
preceding  articles.     The  complete  integral  is 

y  •\-  c  =^  \x^  —  2ax^, 
or 

Uy  +  cy  =  x{x-  sciy (2) 

The  /-discriminant  is 

x(x  -  ay, (3) 

and  the  ^-discriminant  is  ' 

x{x  -  say (4) 

The  system  of  curves  is  the  result  of  moving  the  curve 
J^2  _  ^^^  _  2^)2 

*  If  a  squared  factor  in  the  /-discriminant  satisfies  the  differential  equation, 
the  two  arcs  of  particular  integral  curves  passing  through  (x,  y),  instead  of  crossing 
the  locus  when  (jt,  y)  is  moved  up  to  it,  will  coincide  with  it  in  direction,  as  in  the 
case  of  the  envelope,  Art.  46.  But,  since  /  is  real  on  both  sides  of  the  locus, 
the  arcs  reappear  upon  the  other  side  of  the  locus  when  {x,  y)  is  moved  across  it. 
This  implies  that  they  coincide  with  the  locus  when  {x,  y)  is  upon  it.  Hence,  in  this 
case,  the  squared  factor  appears  also  in  the  ^-discriminant,  and  represents  a  particu- 
lar integral. 


§  v.]  TAC-LOCI  AND  NODE-LOCI.  49 

in  the  direction  of  the  axis  of  y.  This  curve  touches  the  axis 
of  y  at  the  origin,  has  a  node  at  the  point  (3^,  o),  and,  between 
these  points,  consists  of  a  loop  in  which  the  tangents  at  the 
two  points  where  x  ^=^  a  are  parallel  to  one  another.  Accord- 
ingly the  factor  x,  which  is  common  to  both  discriminants  (3) 
and  (4),  indicates  the  envelope  x  =■  o\  x  —  a  =  o  is  a,  tac- 
locus,  and  x  —  $a  :=  o  is  a.  node-locus.  m 

53.  Two  values  of  c  become  equal,  in  other  words,  the  ^-dis- 
criminant vanishes,  whenever  the  point  (x,  y)  is  at  the  ultimate 
intersection  of  consecutive  curves  of  the  system  represented 
by  the  ^-equation.  Suppose  this  equation  to  represent  a  curve 
having,  for  all  values  of  c*  one  or  more  nodes  or  cusps. 
Considering  the  intersections  of  two  neighboring  curves  of 
the  system,  it  is  evident  that  there  are  two  intersections  in  the 
neighborhood  of  each  node,  and  that  these  ultimately  coincide 
with  the  node.  Again,  there  are  three  (all  of  which  may  be  real) 
which  ultimately  coincide  with  each  cusp.  Now,  the  ^-discrimi- 
nant gives  the  complete  locus  of  the  ultimate  intersections  :  it 
therefore  includes  the  node-locus  repeated  twice,  and  the  cusp- 
locus  repeated  three  times ;  that  is  to  say,  the  discriminant 
contains  the  factor  indicating  a  node-locus  as  a  squared  factor, 
and  it  contains  the  factor  indicating  a  cusp-locus  as  a  cubed 
factor,  as  illustrated  in  the  example  of  Art.  49,  where  the  factor 
x^  occurs  in  the  r-discriminant,  while  the  first  power  only  of  x 
occurs  in  the  /-discriminant. 

54.  A  decomposable  differential  equation  of  the  second 
degree  has  no  singular  solution  :   for  the  discriminant  is  the 

*  If,  for  a  particular  value  of  c,  a  node  occurs  at  the  point  {x,  y),  there  are  no 
intersections  of  consecutive  curves  in  its  neighborhood,  the  point  does  not  cause 
the  f-discriminant  to  vanish,  and  there  are  for  it  the  proper  number  of  values  for  r, 
and  therefore  one  too  many  values  of  /.  Hence,  at  such  a  point,  the  /-equation 
vanishes  identically  irrespective  of  the  value  of  / ;  that  is  to  say,  all  its  coefficients 
vanish.     (See  Cayley,  Messenger  of  Mathematics,  New  Series,  vol.  ii.  p.  lo.) 

If  a  point  cause  both  the  /-equation  and  the  ^-equation  to  vanish  identically,  it 
will  be  a  fixed  intersection  of  the  curves  of  the  system. 


so  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.      [Art.  54. 

square  of  the  difference  between  the  roots  ;  hence,  if  the  roots 
are  rational,  it  is  the  square  of  a  rational  function.  The  systems 
representing  the  two  complete  integrals,  in  this  case,  are  non- 
intersecting  systems  ;  and  the  discriminant  vanishes  only  at  the 
tac-locus,  at  every  point  of  which  a  curve  of  one  system  touches 
a  curve  of  the  other  system.  Thus,  in  equation  (i),  Art.  38,  the 
discriminant  is 

{x  -f  yY  -  Axy  -  {x  -  yY, 

the  square  of  a  rational  function  ;  and  the  line  x  =z  y  is  3.  tac- 
locus  at  every  point  of  which  one  of  the  parabolas  represented 
by  equation  (4)  touches  one  of  the  exponential  curves  repre- 
sented by  equation  (5).* 


Examples  V. 

Solve  the  following  equations,  finding  the  singular  solutions, 
when  they  exist,  as  well  as  the  complete  integrals  :  — 


©■- 


Y  =  0,  y  =  ce^'^y     and     y  =  C^-«-^. 


2.  /(/  -  7)  =  x{x  -f  ;;), 

2^  +  ^  =  ^i    ^nd     y  -\-  X  +  1  =  CV^. 

3.  {x^  -}-  i)/2  =  I,  c^e^'y  —  2cxey  =  i. 


*  In  like  manner,  the  discriminant  of  a  decomposable  ^-equation  gives  a  node- 
locus.  But  it  is  to  be  noticed  that  there  is  no  propriety  in  combining  the  two 
integrals  of  a  decomposable  /-equation.  Thus,  if  we  combine  equations  (4)  and  (5) 
of  Art.  38,  assuming  C  and  c  to  be  identical,  we  associate  each  curve  of  one  system 
with  a  particular  curve  of  the  other  system.  But  if,  before  doing  this,  we  change 
the  form  of  one  of  the  integrals  (by  introducing  a  new  constant  f{c),  as  explained 
in  Art.  30),  we  associate  the  curves  differently,  and  obtain  a  new  result,  equally 
entitled  to  be  considered  the  integral  of  the  given  decomposable  differential 
equation. 


§  v.]  EXAMPLES,  51 


cev  =  x^'^. 


IdyV  _a^  ^  ^ 
\dx)        x^ 

5.     y-'f^  4a^,  y^  =  c  ±  ^ax. 

6-     /""  —  5/  +  6  =  o,  y  =  2X  +  c,     and     j'  =  3;^  4-  C 

^-  (iy-f=°'  (j'-.)==4-. 

8.     ^2^2  4-  Z^yP  +  2j'2  =  0,  :ry  =  Cj     and     ^jc^^  =  C 

9.         /3     _|-     2Ji:/2    y2p2     _     2J|f>'2^    =     o, 

y  —  ^}     y  +  x^  =  c,    and    x^'  +  i  +  ^j  =  o. 

10.  p^  —  (xi^  +  xy  ■}•  y^)/>^  +  ^^^(^^  +  xy  +  _y^)/  —  x^y^  =  o, 

rj'  =  e^^^,     cy  =  I  -\-  xy,     and     3jv  =  ^3  ^  ^, 

11.  /^  4-  2/jvcot^  =  y^j  j(i  ±  cos^)  =  r. 

12.  (  — )    —  ax^  =  o,  2K(y  —  cY  =  4ax^, 
\dxl 

13.  X  4-  j(r/2  _  i^  y  _  ^^^  _  ^2^  _|_  sin-'y/jc  4-  c. 

14.  p\x-  +  1)3  =  I,  (^  _  ^)^  =  _^. 

x^  -\-  \ 

15.  j;  =  (^  4-  i)/^, 

^2   4-    2C{X  4-    I    4-   j^)  4-  (^  4-    I    _   ji;)2   =   o. 

16.  jj//^  4-  2^/  —  ji;  =  o,  j2  _  2j:jt:  ^  ^2, 

17.  3^/2  —  dyp  4-  ^  4-  2>'  =  o,  <;2  ^  ^^^  _  2^^  _l_  ^2  _  q^ 

18.  yp  -ir  nx=^  sj{y'^  +  nx^)sj{i  4-  p')j 


52  EQUATIONS  NOT  OF  THE  FIRST  DEGREE.      [Art.  54 


19.     oc^p^  —  2xyp  -\'  y^  —  x^f  -f  ;d,  2C^  =  c^e^^  —  e-*, 

X 


20.  3/*j^  —  2xyp  4-  4;;^  —  :«:»  =  o, 

x^  ■\-  y^  —  A,cx  4-  3^'  =  o. 

21.  jv»(i  +  /»)  =  «H^  +  yp)\ 

{x  -\-  cy  =  {n"  -  i)y^  +  «'^'. 


VI. 

Solution  by  Differentiation. 

55.  The  differentiation  of  a  differential  equation  of  the  first 
order  gives  rise  to  an  equation  of  the  second  order ;  but,  in  the 
cases  now  to  be  considered,  the  result  may  be  regarded  as  an 
equation  of  the  first  order,  and  its  integral  used  in  determining 
that  of  the  given  equation. 

Let  the  given  equation  be  solved  for  j/,  that  is  to  say,  put 
in  the  form 

y  =  f{x,p)) (i) 

then  the  result  of  differentiation  will  be  of  the  form 

which  is  of  the  second  order  as  regards  y,  but,  not  containing 
y  explicitly,  is  an  equation  of  the  first  order  between  x  and  /. 
If,  now,  we  can  integrate  this  equation,  we  shall  have  a  relation 
between  x,  /,  and  an  arbitrary  constant.     The  result  of  elimi- 


§  VI.]  SOLUTION  BY  DIFFERENTIATION.  53 

_ 

nating  /  between  this  equation  and  equation  (i)  will  therefore 
be  a  relation  between  x,  y,  and  an  arbitrary  constant ;  hence  it 
will  be  the  complete  integral  required. 
56.  For  example,  given  the  equation 

-^  ■\-  2xy  =  x^  -^  y^  \ 
ax 

solving  for  7,  we  have 

y  =  x-^s^P', (1) 

and,  differentiating, 

p  =  ^  +  \f (2) 

2^p  ax 
Separating  the  variables  x  and  /,  we  have 


dx  = 


di_ 


and,  integrating. 


2>jp{p  -  I)  ' 


or 


^  ^^:^2^ (  ) 

Finally,  eliminating  /  between  equations  (i)  and  (3),  we  have 
the  complete  integral 

C  +  e^^ 


y  =2  X  + 


C  —  e^^' 


57.  In  attempting  this  mode  of  solution,  it  will  sometimes  be 

more  advantageous  to  treat  y  as  the  independent  variable,  and 

dx 
putting  /'  for  — -,  to  derive  a  differential  equation  involving  y 
ay 

and  /'.      In  either  case,  the  success  of  the  method  depends 

upon  our  ability  to  integrate  the  derived  equation.     The  princi- 


54  EQUATIONS  NOT  OF  THE  FIRST  DEGREE.      [Art.  57. 

pal  cases  in  which  this  can  be  effected  are  those  in  which  one 
of  the  variables  is  absent  and  those  in  which  both  variables 
occur  only  in  the  first  degree. 

It  should  be  noticed  that  the  final  elimination  of  p  is 
frequently  inconvenient,  or  even  impracticable ;  but,  when  this 
is  the  case,  we  may  express  x  and  y  in  terms  of  /  which  then 
serves  as  an  auxiliary  variable. 


Equations  from  which  One  of  the  Variables  is  Absent. 

58.  If  an  equation  of  the  first  order  in  which  x  does  not 
occur  explicitly  can  be  solved  for  /,  it  takes  the  directly  inte- 
grable  form 

f  =  /w w 

y  being  treated  as  the  independent  variable.     Otherwise  let  it 
be  solved  for  y ;  thus, 

J=  </>(/); (2) 

differentiating, 

/  =  *'(/)£, C3) 

in  which  the  variables  x  and  /  can  be  separated. 

In  like  manner,  an  equation  not  containing  y^  if  not  directly 
integrable,  should  be  put  in  the  form 

Differentiating  with  respect  to  y,  we  have 

in  which  the  variables  y  and  /  can  be  separated. 


§  VI.]  ONE    VARIABLE  ABSENT.  55 

59.  As  an  example,  let  us  take  the  equation 

7  =  /^  +  1/3 (i) 

We  have,  by  differentiation, 

/  =  (2/  +  2/^)^, (2) 

which  implies  either  that 

/  =  o, (3) 

or  else  that 

dx  ^  {2  -{■  2p)dp (4) 

Eliminating  /  from  equation  (i)  by  means  of  the  first  of  these, 
which  is  not  a  differential  equation  for/,  we  obtain  the  solution 

*  7  =  o. (5) 

which  does  not  contain  an  arbitrary  constant.    But,  integrating 
equation  (4),  we  have 

^4-^=2/   +   /% 

or 

/  =  —  I  -f  ^{x  -f  c)\ 

and,  employing  this  result  to  eliminate  /  from  equation  (i),  we 
obtain 

or,  rationalizing, 

{x^-  y  ^-  c-\y  =  %{x^  cy (6) 

This  equation  contains  an  arbitrary  constant,  and  is  the  com- 
plete integral. 

Equation  (5),  not  being  a  particular  case  of  equation  (6),  is 
a  singular  solution. 


$6  EQUATIONS  NOT  OF  THE  FIRST  DEGREE.      [Art.  6o. 

60.  With  respect  to  an  equation  of  the  form 

>'  =  </>(/), (i) 

y  =  <^(o), (2) 


it  may  be  noticed  that 


(which,  since  <^  is  not  necessarily  one-valued,  may  include 
several  equations)  is  always  a  solution,  for  it  gives,  by  differ- 
entiation, /  =  o,  and  thus  satisfies  equation  (i).  The  reason 
of  this  is  readily  seen,  for  the  complete  integral  is  capable  of 
expression  in  the  form 

X  =  ^{y)  -\r  c, (3) 

which  is  the  form  it  would  take  if  derived  by  direct  integration 
from  the  form  (i).  Art.  58  ;  it  therefore  represents  the  system 
of  curves  which  results  from  moving  the  curve 

in  the  direction  of  the  axis  of  x.  If  this  curve  contains  points 
at  which  /  =  o,  it  is  evident  that  the  locus  of  these  points,  or 
y  =  </)(o),  is  an  envelope  ;  that  is,  y  =  <^(o)  is  a  singular  solu- 
tion.* But,  if  the  point  for  which  /  =  o  is  at  an'  infinite 
distance,  y  =  </>(o)  will  be  the  particular  integral  corresponding 
to  ^  =  00  when  the  integral  is  written  in  the  form  (3).     For 

*  If  the  /-discriminant  were  formed,  in  this  case,  by  the  general  method  (see 
Art.  48),  we  should  apparently  have  <p'(p)  =  o  as  the  condition  satisfied  alike  by  a 
singular  solution,  a  cusp-locus,  and  a  tac-locus.  But  it  is  to  be  noticed,  that,  when 
0{p)  is  not  a  one-valued  function,  the  method  may  fail  to  detect  a  case  of  equal 
roots.     In  fact  it  is  evident,  from  equation  (3),  Art.  58,  that,  if  ^'(/)  =  o,  we  must 

have  -^  or  ^  infinite,  which  indicates  a  cusp,  except  when  /  =  o,  which,  as  we 

have  seen  above,  gives  a  singular  solution.  Thus,  a  tac-locus  does  not  satisfy 
<^'{p)  =  o.  In  the  example  of  Art.  59,  the  roots  of  ^'{p)  =  o  are  o  and  —  i, 
y  =  <p{o)  being  the  envelope,  while  j  =  ^(  — i)  =  ^  is  a  cusp-locus. 


§  VI.]  HOMOGENEOUS  EQUATIONS.  57 

example,  the  equation  y  =.  p  \s>  satisfied  by  j  =  o.  This  is,  of 
course,  not  a  singular  solution  ;  but  the  complete  integral  is 
log  J  z^  X  -\-  c  or  y  ^=z  Ce^,  and  j/  =  o  is  the  particular  integral 
corresponding  to  ^  =  —  00  in  the  first  form,  or  to  6"  =  o  in  the 
second. 

Homogeneous  Equations. 

61.  When  a  homogeneous  equation  which  is  not  of  the  first 
degree  can  be  solved  for  /,  it  takes  the  form 

dx         \x) 
considered  in  Art.  20.     Otherwise  it  should  be  put  in  the  form 

or 

y  =  x<i>{p) (i) 

Differentiating, 

P  =  HP)  +  ^*'(/)f » (2) 

in  which  the  variables  can  be  separated. 

62.  If  /i  is  a  root  of  the  equation  /  =  <^(/), 

y  =  p^x 

is  always  a  solution  of  equation  (i) ;  for  it  gives,  by  differentia- 
tion, /  =  /„  and  substituting  these  values  in  equation  (i),  we 
have 

p,x  =  X(f>{p,), 

which  is  satisfied  by  the  hypothesis. 

It  was  shown  in  Art.  22  that  the  complete  integral,  in  this 
case,  represents  a  system  of  similar  curves  with  the  origin  as 


58  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.      [Art.  62. 

the  centre  of  similitude.  It  is  hence  evident  that  the  tangents 
from  the  origin  to  any  curve  of  the  system  will,  if  the  points 
of  contact  be  at  a  finite  distance,  constitute  the  envelope  of  the 
system  ;  but,  if  the  points  of  contact  be  at  an  infinite  distance, 
they  will  be  asymptotes  to  the  system.     In  either  case,  they 

y 

will  be  the  loci  of  the  points  for  which  /  =  -  in  the  differen- 
tial equation  (i),  that  is  to  say,  for  which  p  —  <^(/)  ;  but,  in 
the  first  case,  their  equations  will  be  singular  solutions  ;  *  and, 
in  the  second  case,  they  will  constitute  the  particular  integral 
corresponding  to  ^  =  o  when  the  complete  integral  is  written 
in  the  homogeneous  form,  as  in  Art.  22. 

Equation  of  the  First  Degree  in  x  and  y. 

63.   The  equation  of   the  first  degree  in  x  and  y  may  be 
written  in  the  form 

y  =  x<i>{p)  +  f{p) (i) 

Differentiating,  we  have 

/=,/.(/)  +^.^'(/)2  +  /'(/)£,.      ...      (2) 

or 

dp       p-  (/>(/)      ^  /-</>(/)'    .     .     .     .     u; 

which  is  a  linear  equation  for  x  regarded  as  a  function  of  /. 
The  integral  gives  ;ir  as  a  function  of  / ;  the  elimination  of 
/  is  often  impracticable,  but,  in  that  case,  substituting  the 
value  of  X  in  equation  (i),  we  have  x  and  y  expressed  in 
terms  of  /  as  an  auxiliary  variable. 

*  In  this  case  also,  <^\p)  —  o  determines  cusp-loci,  but  fails  to  detect  a  tac- 
locus.     See  the  preceding  foot-note. 


§  VI.]  CLAIRAUrS  EQUATION.  59 

Clairaufs  Equation, 

64.  The  equation 

y  =  px^f{p), ;     (i) 

which  is  a  special  case  of  equation  (i)  of  the  preceding  article, 
is  known  as  Clairaut's  equation.     The  result  of  differentiation 

^  =  ^  +  .|  +  /(,)|. 


or 

"'  '^  o. 


[.  +  r(/)]| 


This  equation  is  satisfied  either  by  putting 

^  +  /(/)  =  0, (3) 

or  by  putting 

^  =  o. (4) 

ax 

Equation  (3)  gives,  by  the  elimination  of  /  from  (i),  a  singular 
solution  ;  and  equation  (4)  gives,  by  integration, 

whence,  from  (i), 

y  ^  ex  -^  f{c). (5) 

This  is  the  complete  integral,  as  is  verified  at  sight,  since 
/  =  ^  is  the  result  of  its  differentiation. 

65.  The  complete  integral,  in  this  case,  represents  a  system 
of  straight  lines,  and  the  singular  solution  a  curve  to  which 
these  lines  are  tangent.  An  example  has  already  been  noticed 
in  Art.  45.     Conversely,  every  system  of  straight  lines  repre- 


6o  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.      [Art.  65. 

sented  by  a  general  equation  containing  one  arbitrary  parameter 
gives  rise  to  a  differential  equation  in  Clairaut's  form,  having, 
for  its  singular  solution,  the  equation  of  the  curve  to  which  the 
system  is  tangent.  We  have  only  to  write  the  equation  in  the 
form  (5),  and  to  substitute  /  for  the  symbol  denoting  the  param- 
eter.    For  example,  the  equation  of  the  tangents  to  the  circle 

x'  -\-  y^  =  a?- 
is 

y  =  mx  4-  «V(i  +  ^«'); 

hence  the  differential  equation  is 

y^  px^  aslii  -f  /^); 
or,  rationalizing, 

{x^  —  a^)p^  —  2xyp  +  >'2  —  a=  =  o. 

Accordingly  the  condition  of  equal  roots  is  found  to  be 
xy  —  (x^  —  a"")  (y   —  a^)  =  o,  or  x""  -]-  j/^  =  a^. 

66.  If  we  form  the  condition  for  equal  roots  in  equation  (i). 
Art.  64,  by  the  general  method  mentioned  in  Art.  48,  we  have 
to  eliminate  p  from  equation  (i)  by  means  of  its  derivative 
with  respect  to  / ;  namely, 

o  =  x±f\p), 


tft] 


which  is  identical  with  equatlBfc).  In  fact,  it  is  obvious  that 
the  condition  should  be  the^Rie  ;  for,  since  the  complete 
integral  represents  straight  lines,  there  can  be  neither  cusp- 
locus  nor  tac-locus.  Precisely  the  same  condition  expresses 
the  equality  of  roots  in  the  ^-equation,  a  node-locus  being  also 
impossible. 


§  VI.]  REDUCTION  TO    CLAIRAUT'S  FORM.  6l 

67.  A  differential  equation  may  be  reducible  to  Clairaut's 
form  by  a  more  or  less  obvious  transformation.  For  example, 
given  the  equation 


j,_,^|  +  «^gJ  =  o; 


since  d{y^)  =  ^ydy,  if  we  multiply  through  by  y,  y'^  may  be 
made  the  dependent  variable  ;  thus, 


f-,^l§L  +  aim  =  o, 
ax  \  ax  I 


or,  putting  y'^  =  v^ 
hence  the  integral  is 


dx      ^\dx) 


y^  =  ex  —  lac^. 


c 


Examples  VI. 
Solve  the  following  differential  equations  :  — 
I.     ^  =  —xp  +  x^p^,  y  =z  -  -\-  c^ 

X 

singular  solution,  i  +  ^x'^y  —  o. 


2.    xp^  —  2yp  -{-  ax  =  o,  2y  —  cx^  -\-^ . 

c 


9 

y  =  T- 


singular  solution,  y''  =  ax- 
S.    X  -{-  py{2p'  +  3)  =  o, 

c  ^       cp{2p-  +  3) 

(I  +  /O^'  ^       (I  +  p')^ 


4.       y  =  ^^P'  X  ^   ""^^   -    ^^    +   C 


62  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.      [Art.  67. 


5.     .Y  +  J//  =  tf/», 


^=^(x  :^^^i^-^^^Qg[/  +  v/(i  ^-/^)]^ 


6.    ;'  =  (I  +  /)^  +  /^, 


(^  =  2(1  —  /)  +  ^^-i*, 
b  =  2  -  /^  +  r^->(i  +  /). 


X  =  a\og  [ay  +  yl{a'  +  J^  —  1)] 

4-  \og\_y  -  sl{a^  +  J*  -  i)]  4-  ^. 

8.  2>'  =  ;r/  +  2> 

a^c^  —  i2acxy  +  ^cy"^  —  \2x^f-  +  i6^^3  =  o. 

9.  ^  =  «/  +  ^/2, 

X  =  a\ogl\J{a^  +  4/^;^)  -  «]  +  v/(«'  +  4^J')  +  ^. 

10.  «'_);/*  —  4jc^  +  ^  :=  o, 

11.  ^  =  ^/  +  V(3^  +  «^/^),  ^  =  ^^  +  v/(^^  +  «V^), 

singular  solution, }-  ^  =  i . 

12.         (l     +     ^0/^     —     2^_)^/    4-    j;2    _     J     _    Q^  jj,    _    ^;^;    ^    y/(l     _     ^2). 

13.  y  =  ^(jc  —  <5)  4-  -,  singular  solution,  y^  =  4<3;(a:  —  ^). 

14.  «^;/>^  4-  (2^  —  d)J>  —  y  =  o,        ac^  +  c{2x  —  d)  —  y  =  o. 
V  15.     fi  -^"  =  e-y  +  e—-(£\\         ^y  =  ce^  +  sj{i  -\-  c^). 

16.  a:2(;;  -  px)  =  :>;/*,  ^2  =  ^^  4-  c\ 

17.  <?3jr(^  —  i)  4-  /3^>'  =  0,  eY  z=  ce^  ^  c^^ 


§  VIL]  EXAMPLES.  63 

18.  {af-  —  b)xy  -\-  {bx"  —  ay"-  +  c)p  =  o, 

Cc 


f  ^Cx"  -^r 


b  ^  aC 


'9-  f-^  =  4|-f^)'  y^c^^^m. 

20.  /3  _  ^xyp  +  8y2  =  o,  y  =  c(x  —  cy. 

21.  x'^p^  —  2{xy  —  2)p  +  y^  s=  o^  {y  —  ^^Y  +  4^  =  o. 

22.  y  •=  2px  +  y'^p^j  y^  z=  ex  -\-  J^3. 

23.  (/^  —  y)  {py  +  x)  ==  h^py  f  ^  CSC"  ^  — 


c  +  I 


VII. 

Geometrical  Applications. 

68.  The  properties  of  a  curve  are  frequently  expressed  by 
means  of  such  magnitudes  as  the  subtangent,  the  subnormal, 
the  perpendicular  from  the  origin  upon  the  tangent,  etc.,  the 
general  expressions  for  which  involve  the  coordinates  of  a 
point  upon  the  curve  together  with  the  value  of  the  derivative 
at  that  point.  Hence  the  analytical  expression  of  such  a 
property,  or,  indeed,  of  any  property  which  depends  upon  the 
tangents  to  the  curve,  gives  rise  to  a  differential  equation. 
Again,  a  property  relating  to  an  area  or  volume  connected  with 
a  curve,  or  to  the  length  of  an  arc  of  the  curve,  is  expressed  by 
a  differential  equation.  Hence  the  problem  to  determine  the 
curve  having  a  given  property  resolves  itself  into  the  solution 


64  GEOMETRICAL   APPLICATIONS.  [Art.  68. 

of  a  differential  equation.  For  example,  the  expression  for  the 
subnormal  is  yp ;  hence,  to  determine  the  curve  whose  sub- 
normal is  constant  and  equal  to  a,  we  have  only  to  solve  the 

differential  equation 

dy 

The  integral  of  this  equation  is 

therefore  the  curve  having  the  given  property  is  the  parabola 
wlttjee  parameter  is  2a,  and  whose  axis  is  the  axis  of  ;r,  the 
position  of  the  vertex  being  indeterminate. 

69.  The  given  property  is,  in  some  cases,  expressed  in  polar 
coordinates.  Thus,  let  it  be  required  to  determine  the  curve 
in  which  the  angle  between  the  radius-vector  and  the  tangent 
is  7t  times  the  vectorial  angle.  Using  the  expression  for  the 
trigonometric  tangent  of  the  angle  first  mentioned,  the  prop- 
erty is  expressed  by  the  equation 

—  =  ia.nn6, 
dr 


or 


dr       cos  nQdB 


Integrating, 


r  sin  nB 

log  r  ==  -  log  sin  nO  +  C, 
which  may  be  written  in  the  form 

The  mode  in  which  the  constant  enters  shows,  as  might  have 
been  anticipated,  that  the  several  curves  which  have  the  prop- 
erty are  simply  similar  curves  similarly  situated  with  respect 
to  the  pole  ;  thus,  when  n  =  i,  they  are  the  circles  which 
touch  the  initial  line  at  the  pole. 


§  VIL]  POLAR   COORDINATES.  65 

70.  As  a  further  illustration,  let  us  consider  the  curve 
traced  by  a  point  carried  by  a  curve  which  rolls  upon  a  fixed 
straight  line.  By  the  principle  of  the  instantaneous  centre, 
the  straight  line  joining  the  carried  point  with  the  point  of 
contact  of  the  curve  with  the  fixed  line  is  always  normal  to 
the  path  of  the  carried  point.  Considering  the  carried  point 
as  a  pole,  this  line  is  a  radius-vector  of  the  given  curve,  and 
the  perpendicular  from  the  carried  point  to  the  fixed  line  is  the 
perpendicular  from  the  pole  upon  a  tangent.  Denoting  these 
lines  by  r^  and  /,  respectively,  the  nature  of  the  given  curve 
determines  a  relation  between  r,  and  /,.  But,  taking  the  fixed 
line  as  the  axis  of  x,  /,  is  an  ordinate  of  the  required  curve, 
and  r,  is  the  part  of  the  normal  intercepted  between  the  point 
of  contact  and  the  axis  of  x,  the  expression  for  which  is 
ysj^i  +  p^).  The  relation  between  /,  and  r,  then  at  once  gives 
the  differential  equation. 

For  example,  let  the  parabola  y^  =  d^ax  roll  upon  a  straight 
line,  and  let  it  be  required  to  determine  the  curve  traced  by 
the  focus.     The  relation  between  p^  and  r„  in  this  case,  is 

therefore  the  differential  equation  is 

f  =  aysl{i  +  f), 


or,  solving  for  /, 


dy  _  yj{y^  —  a^) 
dx  a 


Let  us  take  as  the  origin  the  point  of  the  fixed  line  on  which 
the  vertex  of  the  parabola  falls  in  the  rolling  motion.  This  deter 
mines  the  constant  of  integration  by  the  condition  that  ^  =  o 
when  ^  =  o,  that  is  to  say,  when  y  =  a.     Integrating,  we  have 


f        ^y       =  f  ^ 


66  GEOMETRICAL   APPLICATIONS.  [Art.  70. 

or 

y  +  ^lif-a^)  =  f . 
^  a  a 

which  may  be  reduced  to  the  form 

_y  =  ^M  +  1?"^     =  a  cosh-. 

The  curve  is  the  catenary. 

71.  In  another  class  of  examples,  the  curve  required  is  the 
singular  solution  of  a  differential  equation.  It  is,  in  this  case, 
frequently  possible  to  write  the  complete  integral  at  once,  and 
to  derive  the  singular  solution  from  it  instead  of  forming  the 

ydifferential  equation.  For  example,  required  the  curve  such 
that  the  sum  of  the  intercepts  of  its  tangents  upon  the  axes 

is  constant  and  equal  to  a.  The  equation  of  the  curve  is 
the  singular  solution  of  the  equation  whose  complete  integral 
represents  the  system  of  lines  having  the  property  mentioned. 
The  general  equation  of  this  system  is 

X  y 

7  +  -r=r-c  =  ^> 
in  which  c  is  the  arbitrary  parameter.     Writing  it  in  the  form 

c'^  +  c{y  ^  X  —  a)  -{-  ax  =  o, 
the  condition  of  equal  roots  is 

(y  —  X  —  ay  —  ^ax  =  o, 

or 

(y  —  xY  —  2a{x  -\-  y)  -\-  a^  =  o, 

which  is  the  equation  of  the  required  curve,  and  represents  a 
parabola  touching  the  axes  at  the  points  (a,  o)  and  (o,  a). 


§  VII.]  TRAJECTORIES.  6^ 


Trajectories. 

72.  A  curve  which  cuts  a  system  of  curves  at  a  constant 
angle  is  called  a  trajectory  of  the  system.  The  case  usually 
considered  is  that  of  the  orthogonal  trajectory,  which  cuts  the 
system  of  curves  at  right  angles.  The  differential  equation  of 
the  trajectory  is  readily  derived  from  that  of  the  given  system 
of  curves  ;  for,  at  every  point  of  the  trajectory,  the  value  of  p 
has  a  fixed  relation  to  the  value  of  /  corresponding  to  the  same 
values  of  x  and  y  in  the  equation  of  the  given  system  of  curves. 
Denoting  the  new  value  of  /  by  /',  this  relation  is,  in  the  case 
of  the  orthogonal  trajectories, 

'=-?■ 

If,  then,  we  put in  place  of  ^  in  the  differential  equa- 

dy  dx 

tion  of  the  given  system,  the  result  will  be  the  differential 
equation  of  the  trajectory.  The  complete  integral  of  this  equa- 
tion will  represent  a  system  of  curves,  each  of  which  is  an 
orthogonal  trajectory  of  the  given  system.  Reciprocally,  the 
curves  of  the  given  system  are  the  orthogonal  trajectories  of 
the  new  system. 

73.  For  example,  let  it  be  required  to  determine  the  orthog- 
onal trajectories  of  the  circles  which  pass  through  two  given 
points. 

•  Taking  the  straight  line  which  passes  through  the  two  given 
points  as  the  axis  of  y  and  the  middle  point  as  the  origin,  and 
denoting  the  distance  between  the  points  by  2^,  the  equation 
of  the  given  system  of  circles  is 

oc^  -\-  y'^  -\-  ex  —  b^  =i  Q, (i) 

in  which  c  is  the  arbitrary  parameter.  The  differential  equation 
derived  from  this  primitive  is 

(x^  —  y^  -\-  b'^)dx  +  2xydy  =  o (2) 


68  GEOMETRICAL   APPLICATIONS.  [Art.  yi. 

Substituting for  -/,  we  have 

ay        ax 

(^y^  —  ^'  —  b^)dy  +  2xydx  =  o      .     .     .     .     (3) 

for  the  differential  equation  of  the  trajectories.  This  equation 
is  the  same  as  the  result  of  interchanging  x  and  y  in  equa- 
tion (2),  except  that  the  sign  of  b^  is  changed  ;  its  integral  is 
therefore 

x^  ^  y^  -^Cy  -^  b^  =  o; (4) 

and  the  trajectories  form  a  system  of  circles  having  the  axis 
of  X  as  the  common  radical  axis,  but  intersecting  it  and  each 
other  in  imaginary  points. 

74.  It  is  evident  that  the  differential  equations  of  the  given 
system  and  of  the  orthogonal  trajectories  will  always  be  of  the 
same  degree,  and  that,  wherever  two  values  of  /  become  equal 
in  the  former,  the  corresponding  values  of  /  will  be  equal  in 
the  latter.  Hence  the  loci  of  equal  roots  will  be  the  same 
in  each  case.  Now,  the  trajectories  will  meet  an  envelope  of 
the  given  system  at  right  angles  ;  and,  since  the  values  of  / 
become  imaginary  in  both  equations  as  we  cross  the  envelbpe, 
the  envelope  is  a  cusp-locus  of  the  system  of  trajectories. 
Conversely,  a  cusp-locus  which  is,  at  each  point,  perpendicular 
to  a  curve  of  the  given  system,  becomes  an  envelope  of  the 
system  of  trajectories  ;  but  every  other  cusp-locus  is  also  a 
cusp-locus  of  the  trajectories. 

In  like  manner,  a  tac-locus  of  the  given  system  becomes  a 
tac-locus  of  the  trajectories.*  A  node-locus  gives  rise  to  no 
peculiarity  in  the  system  of  trajectories. 


*  The  case  in  which  the  tangent  curves  of  the  system  cross  the  tac-locus  at 
right  angles  forms  an  exception.  In  this  case,  the  locus  is  itself  one  of  the 
trajectories  ;  and  being  represented,  in  the  common  /-discriminant  of  the  two 
systems,  by  a  squared  factor,  we  have  the  case  considered  in  the  foot-note  on 


§  VII.]  EXAMPLES.  6g 


Examples  VII. 

1.  Determine  the  curve  whose  subtangent  is  n  times  the  abscissa 
of  the  point  of  contact.  jj;«  =  ex. 

2.  Determine  the  curve  whose  subtangent  is  constant,  and  equal 
to  a.  ce^  =  y^. 

3.  Determine  the  curve  in  which  the  angle  between  the  radius- 
vector  and  the  tangent  is  one-half  the  vectorial  angle,     r  =  ^(i  —  cos^). 

4.  Determine  the  curve  in  which  the  subnormal  is  proportional  to 
the  n\h  power  of  the  abscissa.  y^  =  >^.r«+ '  +  c. 

5.  Determine  the  curve  in  which  the  perpendicular  upon  the 
tangent  from  the  foot  of  the  ordinate  of  the  point  of  contact  is  constant 
and  equal  to  a,  determining  the  constant  of  integration  in  such  a 
manner  that  the  curve  shall  cut  the  axis  of  y  at  right  angles. 

The  catenary  y  =  a  cosh-. 
a 

page  48.     For  example,  the  tac-locus  x  z=:am  Art.  52  is  perpendicular  to  the  system 
of  curves  representing  the  complete  integral ;  the  equation  of  the  trajectories  is 

(x  —  aYp^  —  X  =  0,. (i) 

of  which  the  integral  is 

y  +  C  =  2^x  +  \Ja\ogi^^-^^ (2) 

\a  +  Sx 

The  system  is  that  which  results  from  moving  the  curve 


in  the  direction  of  the  axis  of  y.  This  curve  is  symmetrical  to  the  axis  of  x  since 
Sx  admits  of  a  change  of  sign,  and  it  has  a  cusp  at  the  origin,  so  that  the  axis  of  y 
is  a  cusp-locus.  The  line  x  z=  a'xs  an  asymptote  which  is  approached  by  branches 
on  both  sides  of  it ;  and  the  result  of  putting  C  =  00  in  equation  (2)  is,  in  fact,  this 
line,  or  rather  the  line  doubled,  for,  if  C  is  infinite,  we  must,  in  order  to  have  y 
finite,  put  x  z=.  a. 


yo  GEOMETRICAL   APPLICATIONS.  [Art.  74. 

6.  Determine  the  curvef  in  which  the  perpendicular  from  the 
origin  upon  the  tangent  is  equal  to  the  abscissa  of  the  point  of  contact. 

X^   •\-   f   —    2CX. 

7.  Determine  the  curve  such  that  the  area  included  between  the 
curve,  the  axis  of  x^  and  an  ordinate,  is  proportional  to  the  ordinate. 

8.  Determine  the  curve  in  which  the  portion  of  the  axis  of  x 
intercepted  between  the  tangent  and  the  normal  is  constant,  and 
interpret  the  condition  of  equal  roots  for  /. 

2{x  -  c)  =  a\o^{a  ±  sl{a^  ~  4^^')]  T  yj{a^  -  4/)- 

9.  Determine  the  curve  such  that  the  area  between  the  curve,  the 
axis  of  X  and  two  ordinates  is  proportional  to  the  corresponding  arc. 


y  =  cosh 


X 


10.  Determine  the  curve  in  which  the  part  of  the  tangent  inter- 
cepted by  the  axes  is  constant.  x^  -^  yi  —  ah 

11.  Determine  the  curve  in  which  a  and  yS  being  the  intercepts 
upon  the  axes  made  by  the  tangent  ?na  -f-  n/S  is  constant. 

The  parabola  {ny  —  fnx)^  —  2a{ny  +  mx)  ■\-  a^  —  o. 

12.  Determine  the  curve  in  which  the  area  enclosed  between  the 
tangent  and  the  coordinate  axes  is  equal  to  a^. 

The  hyperbola  2xy  =  a^. 

13.  Determine  the  curve  in  which  the  projection  upon  the  axis 
of  y  of  the  perpendicular  from  the  origin  upon  a  tangent  is  constant, 
and  equal  to  a.  The  parabola  x^  =  4^(^  —  y). 

14.  Determine  the  curve  in  which  the  abscissa  is  proportional  to 
the  square  of  the  arc  measured  from  the  origin. 

The  cycloid  v  —  asin~'^  -f  yj{ax  —  x^). 

15.  Determine  the  orthogonal  trajectories  of  the  hyperbolas  xy  =  a. 

The  hyperbolas  x^  —  y^  —  c 


§  VII.]  EXAMPLES.  71 

16.  Determine  the  orthogonal  trajectories  of  the  parabolas  y  =  4^^. 

The  ellipses  2x^  -{-  y^  =  c^. 

1 7.  Determine  the  orthogonal  trajectories  of  the  parabolas  of  the 
;zth  degree  a«-'jv  =  x''.  ny'^  +  x^  =  c"". 

18.  Find  the  orthogonal  trajectories  of  the  confocal  and  coaxial 
parabolas  y^  =  4a{x  +  a).  The  system  is  self-orthogonal. 

19.  Show    generally   that    a    system    of   confocal    conies    is    self- 
orthogonal. 

20.  Find  the  orthogonal  trajectories  of  the  ellipses  —  +  -^  =  i 
when  a  is  constant  and  d  arbitrary.  x^  -^  y^  =  2a^  log  x  -^  c. 

2 1 .  Find  the  orthogonal  trajectories  of  the  cardioids  r  =  ^  ( i  —  cos  6) . 

r  =  c{i  -\-  cosO). 

22.  Determine  the  orthogonal   trajectories  of  the  similar  ellipses 
~  -{-  ^  =  ?i^,  n  being  the  arbitrary  parameter.  y^^  =  cx^^. 

23.  Find  the  orthogonal  trajectories  of  the  ellipses  —^-\-^^—\ 
when-  +  -  =  1.  {xyY^  =  ^^^'+>'^ 

24.  Find    the    orthogonal    trajectories    of   the    system    of  curves 

25.  Find  the  orthogonal  trajectories  of  the  curves  ;-  =  log  tan  Q  -\-  a. 

-  =  sin^'^  +  c. 
r 


72  EQUATIONS  OF   THE  SECOND   ORDER,        [Art.  75. 


CHAPTER    IV. 

EQUATIONS  OF  THE  SECOND  ORDER. 

VIII. 

Successive  Integration. 

75.  We  have  seen,  in  Chapter  I.,  that  the  complete  integral 
of  a  differential  equation  of  the  second  order  must  contain  two 
arbitrary  constants,  and  that  it  is  the  primitive  from  which 
the  given  differential  equation  might  have  been  derived  by 
differentiating  twice  and  using  the  results  to  eliminate  the 
constants.  The  order  in  which  the  differentiations  and  elimi- 
nations take  place  is  evidently  immaterial ;  for,  denoting  the 
constants  by  r,  and  c^^  and  the  first  and  second  derivatives  of 
jv  by  /  and  ^,  all  the  equations  which  can  arise  in  the  process 
form  a  consistent  system  of  relations  between  x^  j/,  c^,  C2,  /, 
and  qy  and  these  are  equivalent  to  three  independent  algebraic 
relations  between  these  six  quantities.  If,  after  differentiating 
the  primitive,  we  eliminate  the  constant  c^,  the  result  will  be  a 
relation  between  Xj  y,  c„  and  /,  that  is  to  say,  a  differential 
equation  of  the  first  order  ;  and,  if  we  further  differentiate 
this  equation,  and  eliminate  r„  the  result  will  be  the  differ- 
ential equation  of  the  second  order.  Now,  regarding  the  latter 
as  given,  the  relation  between  x,  y,  r„  and  /  is  called  a  first 
integral  \  and  the  complete  integral,  or  relation  between  x^  y^ 
c,y  and  ^2,  is  also  the  complete  integral  of  this  first  integral,  c^ 
being  the  constant  introduced  by  the  second  integration. 


§  VIIL]  SUCCESSIVE  INTEGRATION.  73 

76.  As  an  illustration,  let  the  given  equation  be 
If  this  be  multiplied  by  2/,  it  becomes 

^^1+^^!=°' « 

and,  since  this  equation  is  the  result  of  differentiating 

f  ^-  y^  =  c^  ^    \ (3) 

(the  constant,  which  is,  for  convenience,  denoted  by  c^,  dis- 
appearing in  the  differentiation),  equation  (3)  is  a  first  integral 
of  equation  (i).     It  may  be  written 

and  its  integral,  which  is 

sin-'—  =  :^  -f  a, 

or 

y  =  c^m{x  -^  a), (4) 

where  a  is  a  second  constant  of  integration,  is  the  complete 
integral  of  equation  (i).     Expanding  sin  {x  +  a),  and  putting 


I 


A  =  ^COSa,     B  =  ^sina, 
the  complete  integral  may  also  be  written  in  the  form 

y  =  Asinx  ■}-  Bcosx, (5) 

in  which  A  and  B  are  the  two  arbitrary  constants. 


74  EQUATIONS  OF  THE  SECOND   ORDER.        [Art.  7/. 


The  First  Integrals. 

77.  It  is  shown,  in  Arts.  14  and  15,  that  a  differential 
equation  of  the  second  order  represents  a  doubly  infinite 
system  of  curves.  In  fact,  if,  in  the  complete  integral,  we 
attribute  a  fixed  value  to  one  of  the  constants,  we  have  a  singly 
infinite  system  ;  and,  therefore,  corresponding  to  different 
values  of  this  constant,  we  have  an  unlimited  number  of  such 
systems.  For  example,  if,  in  the  complete  integral  (4)  of  the 
preceding  article,  we  regard  f  as  a  fixed  constant,  the  equation 
represents  a  system  of  equal  sinusoids  each  having  the  axis 
of  X  for  its  axis  and  c  for  the  value  of  its  maximum  ordinate, 
but  having  points  of  intersection  with  the  axis  depending  upon 
the  arbitrary  constant  a.  The  first  integral  (3)  is  the  differ- 
ential equation  of  this  system ;  and  equation  (i),  which  does 
not  contain  ^,  represents  all  such  systems  obtained  by  varying 
the  value  of  c. 

On  the  other  hand,  if,  in  equation  (4),  we  regard  a  as  fixed, 
we  have  a  system  of  sinusoids  cutting  the  axis  in  fixed  points, 
but  having  maximunj  ordinates  depending  upon  the  constant  c, 
which  is  now  regarded  as  arbitrary.  If  now  we  differentiate 
equation  (4)  and  eliminate  c^  we  have  the  differential  equation 
of  this  system,  namely, 

7  =  /tan(:c  -f  a), (6) 

which,  being  a  relation  between  x,  j/,  /  and  a  constant,  is 
another  first  integral  of  equation  (i).  The  result  of  eliminating 
/  between  the  first  integrals  (3)  and  (6)  would,  of  course,  be 
the  complete  integral  (4). 

78.  Consider  now  the  form  (5)  of  the  complete  integral.  If 
we  regard  A  as  fixed,  the  singly  infinite  system  represented  is 
one  selected  in  still  another  manner  from  the  doubly  infinite 
system  ;  it  consists,  in  fact,  of  those  members  of  the  doubly 
infinite  system  which  pass  through  the  point  (Jtt,  A).      The 


§  VIIL]  THE  FIRST  INTEGRALS.  75 

differential  equation  of  this  system,  which  is  found  by  differen- 
tiating, and  eliminating  B,  is 

jsinjx:  -f  pco^x  =  Af (7) 

which  is,  accordingly,  another  first  integral  of  equation  (i) 
Again,  regarding  B  as  fixed,  and  eliminating  A  from  equation 
(5),  we  obtain  the  first  integral 

ycosx  —  psinx  =  B (8) 

In  like  manner,  to  every  constant  which  may  be  employed 
as  a  parameter  in  expressing  the  general  equation  of  the  doubly 
infinite  system  of  curves  there  corresponds  a  first  integral  of 
the  differential  equation  of  the  second  order.  Thus,  the 
number  of  first  integrals  is  unlimited. 

79.  If  c,  and  C2  are  two  independent  parameters,  that  is  to 
say,  such  that  one  cannot  be  expressed  in  terms  of  the  other, 
all  the  other  parameters  may  be  expressed  in  terms  of  these 
two.  Accordingly,  the  two  first  integrals  which  correspond  to 
c^  and  <:2,  which  may  be  put  in  the  form 

/i(-^>  y^  P)  =  ^i,     fzk^y  y,  P)  =  ^2» 

may  be  regarded  as  two  independe^it  first  integrals  from  which 
all  the  first  integrals  may  be  derived.  For  example,  if  the  first 
integrals  (7)  and  (8)  of  the  preceding  article  be  regarded  as 
the  two  independent  first  integrals,  equation  (3)  of  Art,  'j6 
may  be  derived  from  them  by  squaring  and  adding,  because 
c^  =  A"-  -\-  B\ 

It  must  be  remembered  that  no  two  first  integrals  are 
independent  when  regarded  as  differential  equations  of  the 
first  order  ;  for  they  must  both  give  rise,  by  differentiation,  to 
the  same  equation  of  the  second  order.  They  are  only  inde- 
pendent in  the  sense  that  the  constants  involved  are  independ- 
ent, so  that  they  may  be  regarded  as  independent  algebraic 


76  EQUATIONS  OF  THE  SECOND   ORDER.        [Art.  79. 

relations  between  the  five  quantities  x^  j,  /,  ^„  and  c^^  from 
which,  by  the  elimination  of  /,  the  relation  between  x,  j,  f„ 
and  fj  can  be  found  independently  of  the  differential  relation 
between  x^  y^  and  p. 

Integrating  Factors. 

80.  If  a  first  integral  of  a  given  differential  equation  of  the 
second  order  be  put  in  the  form  f{xy  y^  p)  z=z  c  and  differen- 
tiated, the  result,  not  containing  c^  will  be  a  relation  between 
X,  y,  py  and  ^,  which  is  satisfied  by  every  set  of  simultaneous 
values  of  these  quantities  which  satisfies  the  given  differential 
equation.  This  result  will  therefore  either  be  the  given  equa- 
tion, or  else  the  product  of  that  equation  by  a  factor  which  does 
not  contain  {/.  In  the  first  case,  the  given  equation  is  said  to 
be  an  exact  differential  equation  ;  in  the  latter,  the  factor  which 
makes  it  exact  is  called  an  integrating  factor.  In  general,  to 
every  first  integral  there  corresponds  an  integrating  factor. 
For  example,  differentiating  equations  (7)  and  (8)  of  Art.  'j'^y 
we  find  the  corresponding  integrating  factors  of  the  equation 

doc^ 

to  be  cos  X  and  sin  x  respectively.  Again,  the  integrating 
factor  /  was  employed,  in  Art.  ^6,  in  finding  the  first  integral 
(3)  by  means  of  which  we  solved  the  equation. 

81.  It  is  to  be  noticed  that  an  exact  equation  formed,  as  in 
the  case  last  mentioned,  by  means  of  an  integrating  factor 
containing  /,  is  really  a  decomposable  equation  consisting  of 
the  given  differential  equation  of  the  second  order  and  the 
differencial  equation  of  the  first  order  which  results  from 
putting  the  integrating  factor  equal  to  zero.  The  exact  differ- 
ential equation  therefore  represents,  in  this  case,  not  only  the 
doubly  infinite  system,  but  also  a  singly  infinite  system  which 
does  not  satisfy  the  given  differential  equation.     This  system 


§  VIII.]  INTEGRATING  FACTORS.  J  J 

consists  of  the  singular  solutions  of  the  several  singly  infinite 
systems  represented  by  the  first  integral  when  different  values 
are  given  to  the  constant  contained  in  it.  For  example,  equa- 
tion (2),  Art.  j6,  is  satisfied  by  j/  =  (7,  which  does  not  satisfy 
equation  (i),  but  is  the  solution  of  /  =  o;  accordingly,  the  first 
integral  (3)  has  the  singular  solutions  y  =:  ±c^  which,  when  c 
is  arbitrary,  form  the  singly  infinite  system  of  straight  lines 
parallel  to  the  axis  of  x.  In  fact,  a  singular  solution  of  a  first 
integral  represents  a  line,  which,  at  each  of  its  points,  touches 
a  particular  curve  of  the  doubly  infinite  system.  The  values 
of  X,  f,  and  /,  for  a  point  moving  in  such  a  line,  are  therefore 
the  same  as  for  a  point  moving  in  a  particular  integral  curve  ; 
but  the  values  of  q  are,  in  general,  different ;  *  hence  such  a 
point  does  not  satisfy  the  given  differential  equation. 


*  The  values  of  ^  will,  however,  be  the  same  if  the  line  in  question  has  at 
every  point  the  same  curvature  as  the  particular  integral  curve  which  it  touches 
at  that  point ;  and  its  equation  will  then  be  a  singular  solution.  The  case  is 
analogous  to  that  of  the  singular  solution  of  an  equation  of  the  first  order ;  the 
given  equation  being  supposed  of  a  degree  higher  than  the  first  in  ^,  and  a 
necessary  (but  not  a  sufficient)  condition  being  that  two  values  of  ^  shall  become 
equal  for  the  values  of  x,  y,  and  /  in  question.  Suppose,  for  example,  the  doubly 
infinite  system  of  curves  represented  by  the  differential  equation  to  consist  of  all 
the  circles  whose  centres  lie  upon  a  fixed  curve.  In  order  to  determine  the 
particular  integrals  which  pass  through  an  assumed  point  [x,  y)  in  the  direction 
determined  by  an  assumed  value  of  /,  we  must  draw  a  straight  line  through  [x,  y) 
perpendicular  to  the  assumed  direction,  the  required  particular  integrals  being 
circles  whose  centres  are  the  points  where  this  line  cuts  the  fixed  curve.  These 
circles  correspond  to  the  several  values  of  q  which  are  consistent  with  the  assumed 
values  of  x,  y,  and  /.  When  the  line  touches  the  fixed  curve,  two  of  the  values  of 
q  are  equal,  and  the  values  of  x,  y,  and  /  satisfy  the  condition  of  equal  roots  in 
the  differential  equation  considered  as  an  equation  for  q.  Consider  now  an  involute 
of  the  fixed  curve  ;  its  normals  touch  the  given  curve  ;  hence  the  values  of  x,  y, 
and  /,  at  any  of  its  points,  satisfy  the  condition  of  equal  roots.  Now,  the  circle 
corresponding  to  the  twofold  value  of  q  is  the  circle  of  curvature  of  the  involute, 
so  that  the  value  of  q  for  a  point  moving  in  the  involute  is  the  same  as  its  value  for 
a  point  moving  in  a  particular  integral  curve,  and  the  equation  of  the  involute  is  a 
singular  solution.  Thus  the  involutes  of  the  fixed  curve  constitute  a  singly  infinite 
system  of  singular  solutions,  and  the  relation  between  x,  y,  and  /,  which  is  satisfied 


yS  EQUATIONS  OF   THE  SECOND   ORDER.         [Art.  ^2. 


Derivation  of  the  Complete  Integral  from  Two  First  Integrals. 

82.  It  sometimes  happens  that  it  is  easier  to  obtain  two 
independent  first  integrals  than  to  effect  the  integration  of  one 
of  the  first  integrals.  The  elimination  of  /  between  the  two 
first  integrals  then  gives  the  complete  integral.  For  example, 
as  an  obvious  extension  of  the  results  obtained  in  Art.  80,  we 
see  that  both  cos  ax  and  sin  ax  are  integrating  factors  of  the 
equation 

and,  since   these   expressions   contain  x  only,  they  are   also 
integrating  factors  of  the  more  general  equation;- 

0  +  '^^^  =  ^ w 

if  X  is  a  function  of  x  only.     Thus,  we  have  the  exact  differ- 
ential equation, 

CQsax—^  +  a^yQ,0'?>ax  —  X  0,0%  ax. 
dx^  ^ 

and  its  integral,  which  is 

cos^jc--^  -f-  ay€vciax  =  \X  co'?>  axdx  ■\-  c^  ,     .     .     (2) 
dx  J 

is  a  first  integral  of  equation  (i).     In  like  manner,  the  integrat- 
ing factor  sin  ax  leads  to  the  first  integral 

^vaax-^  —  ^^cos^;i;  =    X sin ^^if^jc  —  c^.    ,     .     .     (3) 
dx  J 


by  all  the  involutes  (in  other  words,  their  differential  equation)  satisfies  the  con- 
dition of  equal  roots;  that  is  to  say,  it  is  the  result  of  equating  to  zero  the 
discriminant  of  the  ^-equation  or  one  of  its  factors. 


§  VIIL]  ELIMINATION  OF  p  FROM  TWO  FIRST  INTEGRALS.  79 

Eliminating  /  between  equations  (2)  and  (3),  we  have 

ay  =  sin  ax  X  cos  axdx  —  cos  ax  X  sin  axdx  +  c^  sin  ax  +  c^  cos  ax^ 

the  complete  integral  of  equation  (i). 

83.  The  principle  of  this  method  has  already  been  applied 
to  the  solution  of  equations  of  the  first  order  in  Art.  55.  The 
method  there  explained,  in  fact,  consists  in  forming  the  equa- 
tion of  the  second  order  of  which  the  given  equation  is  a  first 
integral,  then  finding  an  independent  first  integral,  and  deriving 
the  complete  integral  by  the  elimination  of  /.  But  it  is  to  be 
noticed  that  the  given  equation,  containing,  as  it  does,  no  arbi- 
trary constant,  is  only  a  particular  case  of  the  first  integral  of 
the  equation  of  the  second  order  corresponding  to  a  particular 
value  of  the  constant  which  should  be  contained  in  it.  Accord- 
ingly, the  final  equation  is  the  result  of  giving  the  same  par- 
ticular value  to  this  constant  in  the  complete  integral  of  the 
equation  of  the  second  order.  For  example,  in  the  solution  of 
Clairaut's  equation.  Art.  64,  the  equation  of  the  second  order 

is  —^  =  o ;  the  first  integral,  of  which  the  given  equation  is  a 

special  case,  is  jj/  -f-  C  ^=^  xp  -\-  f{p)  ;  and  the  complete  inte- 
gral \^  y  -\-  C  =  ex  -^  /(c),  which  represents  all  straight  lines  ; 
whereas  the  required  result  is  the  singly  infinite  system  of 
straight  lines  corresponding  to  6'  =  o.* 

*  In  accordance  with  Art.  81,  it  would  seem  that  a  singular  solution  of  the 
given  equation,  when  it  exists,  could  not  satisfy  the  equation  of  the  second  order, 
and  therefore  must  correspond  to  a  factor  which  divides  out,  just  as  x  -\-  f'[p) 
does  in  the  solution  of  Clairaut's  equation.  This  is  indeed  true  when  the 
singular  solution  belongs  to  the  generalized  first  integral,  as  in  this  case  it  does 
to  jj/  +  C  =.cx  4-/(<r).  But  generally  the  singular  solution  belongs  only  to  the  given 
equation ;  and  there  is  no  reason  why  a  singular  solution  of  a  particular  first 
integral  should  not  satisfy  the  differential  equation  of  the  second  order.  Thus  a 
singular  solution  does  not  generally  present  itself  in  the  process  of  "solution  by 
differentiation,"  as  it  does  in  the  case  of  Clairaut's  equation. 


80  EQUATIONS  OF  THE  SECOND   ORDER.        [Art.  84. 

Exact  Differential  Equations  of  the  Second  Order. 

84.  An  exact  differential  equation  of   the  second  order  is 
the  result  of  differentiating  a  first  integral  in  the  form  . 

Axy  y,  P)  =  c (i) 

Hence  it  will  be  of  the  form 

1  +  1^-^1^  =  °' <^) 

in  which  the  partial  derivatives  -f^,    -f^  and  -^,  are  functions 

ax     ay  dp 

of  x^  y  and  / ;   so  that  the  latter  forms  the  entire  coefficient 

of  q  in  the  equation.     Hence,  if  a  given  equation  of  the  second 

order  is  exact,  we  can,  from  this  coefficient,  find,  by  integration 

with  respect  to  /,  the  form   of   the  function  /  so  far  as   it 

depends  upon  / ;   that   is  to   say,  we  can  find  all  the  terms 

of  the  integral  which  contain  /.      These  terms  being  found, 

their  complete  derivative  must  be  subtracted  from  the  first 

member  of  the  given  differential  equation,  and  the  remainder, 

which  will  be  a  differential  expression  of  the  first  order,  must 

be  examined.     If  this  remainder  is  exact,  the  whole  expression 

is  evidently  exact ;   and  its  integral  is  the  sum  of  the  terms 

already  found  and  the  integral  of  the  remainder. 

85.  As  an  illustration,  let  the  given  equation  be 

The  terms  containing  q  are  (i.  —  ;r^)  -^ ;  and,  integrating  this 

ax 

with  respect  to  /,  we  have  (i  —  x'^)p  for  the  part  of  the  integral* 


§  VIII.]  EXACT  EQUATIONS.  3l 

which  contains  /.     The  complete  derivative  of  this  expression 
is 

^  Ux"  dx' 

and,  subtracting  this  from  the  first  member  of  equation  (i),  we 
have  the  remainder 

x-l  4-^  =  0, 
ax 

which  is  the  derivative  of  xy.      Hence  equation  (i)  is  exact, 

and  its  integral  is 

(i  —  x^)p  -\-  xy  =  c,. (2) 

Again,  if  we  multiply  equation  (i)  by  /,  it  becomes 

(I  -  :r^)/^  -  ^/»  +  ;;/  =  o (3) 

In  this  form,  the  integral  of  the  terms  containing  ^  is  J(i  —  x^)p^, 
of  wliich  the  complete  derivative  is 

(i  -  x^)p^  -  xf, 
dx 

The  remainder,  in  this  case,  is  ypy  which  is  the  exact  derivative 
of  \y^  \  hence  equation  (3)  is  also  exact,  and  its  integral  is 

(I  -  ^^)/^  +  JJ^  =  ^. (4) 

Equations  (2)  and  (4)  are  two  first  integrals  of  equation  (i)  ; 
hence,  eliminating  /,  we  have  the  complete  integral 

c^  —  2c,xy  4.  j>;2  —  ^2(1  —  a:^)  =  o,  .     .     .     .     (5) 

which  represents  a  system  of  conies  having  t^eir  centres  at  the 
origin,  and  touching  the  straight  lines  x  ■=  ±1. 


83  EQUATIONS  OF   THE  SECOND   ORDER.        [Art.  ^6, 


Equations  in  which  y  does  not  occur, 

86.  A  differential  equation  of  the  «th  order  which  does  not 
contain  y  is  equivalent  to  an  equation  of  the  («  —  i)th  order 
for/.  The  value  of  /  as  a  function  of  x  obtained  by  integrating 
this  will  contain  n  —  i  constants ;  and  the  remaining  constant 
will  appear  in  the  final  integration,  which  will  take  the  form 

y  =»  \pdx  4-  C, 

If  the  given  equation  is  of  the  first  degree  with  respect  to  the 
derivatives,  it  will  be  a  linear  equation  because  the  coefficients 
do  not  contain  y.  Thus,  if  the  equation  is  of  the  second  order, 
it  may  be  put  in  the  form 

gn- /(.)!  =  ,(.), 

or 

a  linear  equation  of  the  first  order  for  /.  For  example,  the 
equation 

dx^  dx 

is  equivalent  to 

dx       I  +  x^^  I  +  ^' 


The  integral  of  this  is 


/  =  -«  +         ^' 


and,  integrating  again, 

y  =  c^  -  ax  -\-  ^,log[^  -f  v^(i  +  ^')]. 


§  VIIL]     EQUATIONS  IN  WHICH  X  DOES  NOT  OCCUR.  83 

87.  In  general,  an  equation  of  the  ;2th  order  which  does  not 
contain  y,  and  in  which  the  lowest  derivative  is  of  the  rth 
order,  is  equivalent  to  an  equation  of  the  {71  —  r)th  order  for 
the  determination  of  this  derivative.     For  example, 


is  equivalent  to 
Integrating,  we  have 


d^q 

dx^  ^ 


dx^ 
and,  integrating  twice  more. 


Equations  in  which  x  does  not  occur. 

88.  An  equation  of  the  second  order  in  which  x  does  not 
occur  may  be  reduced  to  an  equation  of  the  first  order  between 
y  and  p  by  putting 


d'^y  _  d^  ^  dp  dy  _     dp 
dx^       dx       dy  dx  dy' 


For  example,  the  equation 

dx 


.djy^/dy\^ 

^  ^x^       \dx)  ^^ 


thus  becomes 

dy 


fP^  ^  P\ (2) 


or 

dp  _  dy  ^ 

p^~y' 


84  EQUATIONS  OF  THE  SECOND   ORDER.        [Art.  88. 


whence 


/     y 


or 

dx  —  -^  -\-  Cidv'y 

y 

and,  integrating  again, 

X  =  \ogy  -{-  c,y  +  c^ (3) 

In  equation  (2),  we  rejected  the  solution  /  =  o,  which 
gives  y  z=:  C  \  but  it  is  to  be  noticed  that  the  equation  is  still 
satisfied  by  /  ==  o  after  the  rejection  of  the  factor  /;  accord- 
ingly, y  =.  C  IS  3.  particular  system  of  integrals  included  in  the 
complete  integral  (3),  as  will  be  seen  by  writing  the  latter  in 
the  form 

y  ==  A  ^  B{x  -  logj), 

and  making  B  =■  o. 

89.  If  the  equation  contains  higher  derivatives,  they  may, 
in  like  manner,  be  expressed  in  terms  of  derivatives  of  /  with 
respect  to  y.     Thus, 

dx^        dxdx^       ^  dyY  dyj  df       ^\dy) 

In  like  manner,  the  expression  for  the  fourth  derivative  may  be 

found  by  applying  the  operation  /  -7  to  this  last  result,  and  so 

ay 

on. 

The  Method  of  Variation  of  Parameters. 

90.  When  the  solution  of  an  equation  in  which  the  second 
member  is  zero  is  known  in  the  form  y  =■  f{x)^  the  more 
general  equation  in  which  the  second  member  is  a  function 
of  X  may  sometimes  be  solved  by  assuming  the  value  of  y  in 


§  VIII.]    THE  METHOD   OF   VARIATION  OF  PARAMETERS.    85 

the  same  form  as  that  which  satisfies  the  simpler  equation, 
except  that  the  constants  or  parameters  in  that  solution  are 
now  assumed  to  be  variables.  By  substituting  for  y  in  the 
given  equation  its  assumed  value,  we  obtain  an  equation  which 
must  be  satisfied  by  these  new  variables.  When  the  given 
equation  is  of  the  first  order,  there  is  but  one  new  variable, 
and  the  method  amounts  merely  to  a  transformation  of  the 
dependent  variable ;  but  when  the  equation  is  of  the  f/th  order, 
the  assumption  involves  ;/  new  variables,  and  we  are  at  liberty 
to  impose  ;/  —  i  other  conditions  upon  them  beside  the  con- 
dition that  the  given  equation  shall  be  satisfied.  The  condi- 
tions which  produce  the  simplest  result  are  that  the  derivatives 
of  J,  of  all  orders  lower  than  the  ;/th,  shall  have  the  same  values 
when  the  parameters  are  variable  as  when  they  are  constant. 
91.  For  example,  given  the  equation 


we  assume 


</^  +  '''-^  =  ^' <'> 


y  =  C^Q.o'?>ax  +  CjSin^^, (2) 

which,  if  C^  and  C^  are  constant,  satisfies  the  equation  when 
X  =  o.  Now,  if  C,  and  {Tj  are  variable,  we  may  assume  this 
value  of  y  to  satisfy  equatioii  (i),  and,  at  the  same  time,  impose 
a  second  condition  upon  the  two  new  variables.  Differentiating, 
we  have 

-^  =  —aCi^max  +  aC2  cos  ax  H cos  ax  -| sm  ax, 

dx  dx  dx 

in  which  the  first  two  terms  form  the  value  of  -^  when  C^  and 

dx 

C2  are  constant.     We  now  assume,  as  the  second  condition 

mentioned  above, 

dCi               ,    dC2  •  /  X 

— -0.0%  ax  H -^Ys\ax  =  0, (3) 

dx  dx 


86  EQUATIONS  OF  THE  SECOND  ORDER.        [Art.  9 1. 


which  makes 


JL  —  —aCy'SiViax  +  aC^  cos  ax, 
dx 


Differentiating  again,  we  have 

-^  =  — a}  C  ^0.0%  ax  —  a^CiSinax  —  a — •'sm^.r  +  a — -cos  ax. 
dx^  dx  dx 


Substituting  in  equation  (i),  we  obtain 

—  a — ^sin^^^c  -f  a — ^cos^jc  —  X    .     ,     .     .     (4) 
dx  dx 

as  the  condition  that  y,  in  equation  (2),  shall  satisfy  the  given 
equation.     Equations  (3)  and  (4)  give,  by  elimination, 


—  a  — -'  =  X  sin  ax,       a  — ?  =  X  cos  ax ; 
dx  dx 


whence 


Ci  =  —  Xsin^;c^:r  4-  Cj,      Cj  =  -\X cos axdx  +  ^2; 
and,  substituting  in  equation  (2), 

y  —  — cos  ax\X  sm  axdx  -\ — sm  ax\X  cos  axdx 

-H  Ci  cos  ax  4-  ^2  sin  ax, 
as  otherwise  found  in  Art.  82. 

The  method  of  variation  of  parameters  is  of  historic  interest 
as  one  of  the  earliest  general  methods  employed.  It  may 
occasionally  be  applied  also  when  the  term  neglected  in  finding 
the  form  to  be  assumed  for  the  value  of  y  is  not  a  mere 
function  of  x ;  but,  for  the  most  part,  examples  which  can 
be  solved  by  it  can  be  more  readily  solved  by  the  methods 
given  in  the  succeeding  chapters. 


§  VIII.]  EXAMPLES.  8y 

Examples  VIII. 
Solve  the  following  differential  equations  :  — 

I.     -j^  =  xe"^,  y  =i  {x  —  2)^-^  -\-c^x  H-^Tj. 

ax'* 


2. 


—^  =  sin3^,      7  =  Jcos'a;  —  -^cos'^  +  c^x^  +  c^x  +  Cy 


dx^ 


4.  Find  a  first  integral  of  yf  =  /(^-^),  (^)'  =  2  \f{x)dx. 

5.  — ^  =~^J£:  +  ^_>'  [3  >  o],  ~ax  ■\-  by  —  Ae"^^  +  Be-"^^, 


d^y 

flt^i:  —  ^j'  =  ^sin^y/3  +  BcQ^xsjb, 


6.     -^  =  «^  —  ^y  f/^  >  ol, 

dx^  ^  ^         -'' 


dx^  >J{2ey  +  c^)  ^  c  ' 

e-y=     V"     ,    or 

c  —  ^ 

2^:''  =  c^%tz'-{\cx  +  C), 
according  as  the  first  constant  of  integration  is  c^,  o,  or  —  ^. 

^      ^^v        I  dy\    , 

»•  ^=(i;  +''  .,.^= cos  (.+..), 


BS  EQUATIONS  OF   THE  SECOND   ORDER.        [Art.  9 1. 


ax*       X  ax 


dx*         ax  a  —  y 


dx*         ax  J 

13.  -^  +  MY  +1  =  0,  ;;  =  logsin(;c  -  a)  +  ^. 

14.  Show  that  y— ;i?  -^  is  an  exact  differential. 

^  -^  dt^  dt*    • 


dx*       x^  X 


16.     (i  -  x")^  -  a:^  =2,     y  =  (sin-^;«:)^  +  ^.sin-^;*;  +  c^. 


17.     (i  -  x^)-f-  +  ^-^  =  ^^, 
ax^  ax 

y  z=  ax  •\-  ^j[sin-^.:v  +  ^^{^  —  •^■^)]  +  ^2' 


-  <'-^->£+'+(iy=°' 


§  VIII.]  EXAMPLES.  89 


22.     ;;(!  -  log J)^  +  (i   4-  log>')(;^j   =  o» 


^3-^£-(i;=>^'o^-' 


log^  =  I  + \ 


log  J  =  c^e^  -\-  c^e-^. 


2d( 


2§.       -— -    +   «    = 


v/(i  +  /^^sin^^)    ,  ,^  . 

u  =  J^-i^ — -^- ^  +  ^cos(^  —  a). 

I  +  ^' 


27.  Determine  the  curve  in  which  the  normal  is  equal  to  the 
radius  of  curvature,  but  in  the  opposite  direction. 

The  catenary  7  =  <rcosh-. 
c 

28.  Determine   the   curve   in  which   the   radius   of  curvature   is 
double  the  normal,  and  in  the  same  direction. 

The  cycloid  ^x  —  <;sin-'^  ~  -^  +  \l{2cy  —  y). 


90  EQUATIONS  OF  THE  SECOND  ORDER.        [Art.  91. 

29.  Determine   the  curve   in  which   the   radius  of  curvature   is 
double  the  normal,  and  in  the  opposite  direction. 

The  parabola  x^  =  ^ci^y  —  c), 

30.  Show  that  the  equation 

dx'  dx       ^\dx) 

can  be  solved  in  the  following  cases  :  (a)  when  P  and  Q  are  functions 
of  X ;  (y8)  when  P  and  Q  are  functions  of  y ;  (y)  when  /*  is  a  func- 
tion of  X  and  Q  a  function  of  y. 

In  the  case  (a),  the  equation  is  of  the  "extended  linear  form," 

Art.  37,  for  -^ ;  in  the  case  (^),  x  does  not  occur,  as  in  Art.  88  ;  and  in 
dx 

the  case  (y),  the  equation  is  exact  when  divided  by  -^. 

dx 


variation   of  parameters,  the  assumed   form  of  -f  being  derived  by 

dx 


In  the  last  case,  the  equation  may  also  be  solved  by  the  method  of 
ation   of  parameters,  the  assumed 
neglecting  the  last  term;   the  result  is 

U^'^dy  =  Ale-i'^'^dx  +  B. 


§  IX.]         PROPERTIES  OF   THE  LINEAR  EQUATION.  9 1 


CHAPTER  V. 

LINEAR  EQUATIONS   WITH   CONSTANT   COEFFICIENTS. 

IX. 

Properties  of  the  Linear  Equation. 

92.  A  linear  differential  equation  is  an  equation  of  the  first 
degree  with  respect  to  y  and  its  derivatives.  The  linear 
equation  of  the  ;^th  order  may  therefore  be  written  in  the  form 

^^^^    ^^  __^  ...  (I) 

in  which  the  coefficients  Po,  Pi  .  .  .  P„  may  either  be  constants 
or  functions  of  x^  and  the  second  member  X  is  generally  a 
function  of  x. 

We  have  occasion  to  consider  solutions  of  linear  equations 
only  in  the  form  y  =  /{x),  and  it  is  convenient  to  call  a  value 
of  J/  in  terms  of  x  which  satisfies  the  equation  an  integral  of 
the  equation.  Thus,  if  7,  is  a  function  of  x,  such  that  y  ^=^  y^ 
satisfies  equation  (i),  we  shall  speak  of  the  function  y^,  rather 
than  of  the  equation  jj/  =  j^^ ,  as  an  integral  of  equation  (i). 

93.  The  solution  of  equation  (i),  whether  the  coefficients  be 
variable  or  constant,  is  intimately  connected  with  that  of 

^"ri  +  ^'^  +  ---  +  ^«^  =  °'    .  .  •  (^) 

dx^  dx*^-^ 

which  differs  from  it  only  in  having  zero  for  its  second  member. 


92   LINEAR  EQUATIONS:   CONSTANT  COEFFICIENTS.  [Art.  93. 

Let  y^  be  an  integral  of  equation  (2) ;  then  6",j„  where  6', 
is  an  arbitrary  constant,  is  also  an  integral.  For,  if  we  put 
y  =  C.y^  in  the  first  member,  the  result  is  the  product  by  Cj  of 
the  result  of  substituting  y  =  y^;  and,  since  the  latter  result 
vanishes,  the  former  will  also  vanish. 

Again,  let  y2  be  another  integral  of  equation  (2),  which  is 
not  of  the  form  C,yi  ;  then  will  ^jj^j  be  an  integral,  and 
^1^1  +  ^'2^2  will  also  be  an  integral.  For  the  result  of  putting 
y  =  C,y,  +  (Tjjz  in  the  first  member  will  be  the  sum  of  the 
results  of  putting  y  =  dyj  and  y  =  Cxyz  respectively,  and 
will  therefore  vanish.  In  like  manner,  if  j„  y^,  yi  -  -  -  yn  are 
n  distinct  integrals  of  equation  (2), 

y  =  C,y,  +  C2>'2  +  . .  .  +  Cnyn      ....     (3) 

will  satisfy  the  equation ;  and,  since  this  expression  contains 
n  arbitrary  constants,  it  will  be  the  complete  integral  of 
equation  (2).  Thus  the  complete  integral  is  known  when  ;/ 
particular  integrals  are  known,  provided  they  are  distinct ;  that 
is  to  say,  such  that  no  one  can  be  expressed  as  a  sum  of 
multiples  of  the  others. 

94.  Now  let  Y  denote  a  particular  integral  of  the  more 
general  equation  (i),  and  let  u  denote  the  second  member  of 
equation  (3),  that  is  to  say,  the  complete  integral  of  equation  (2). 
If  we  substitute 

y  =  Y  +  u (4) 

in  the  first  member  of  equation  (i),  the  result  will  be  the  sum 
of  the  results  of  putting  y  =.  Y,  and  y  =^  u  respectively.  The 
first  of  these  results  will  be  X  because  F  satisfies  equation  (i), 
the  second  result  will  be  zero  because  u  satisfies  equation  (2) ; 
hence  the  entire  result  will  be  X,  and  equation  (4)  is  an  integral 
of  equation  (i).  Moreover,  it  is  the  complete  integral  because 
//  contains  n  arbitrary  constants.     Thus  the  complete  integral 


§  IX.]         PROPERTIES  OF  THE  LINEAR  EQUATION.  93 

of  equation  (i)  is  known  when  any  one  particular  integral  is 
known,  together  with  the  complete  integral  of  equation  (2). 

In  equation  (4),  Y  is  called  the  particular  integral,  and  it 
is  called  the  complementary  function.  The  particular  integral 
contains  no  arbitrary  constants,  and  any  two  particular  integrals 
may  differ  by  any  multiples  of  one  or  more  terms  belonging  to 
the  complementary  function. 


Linear  Equations  with  Constant  Coefficients  and  Second  Member  Zero. 
95.  In  the  equation 

dx*"  dx"^-^  dx 

in  which  the  coefficients  A^,  A,  .  ,  .  A„  are  constants,  let  us 

substitute  y  =■  e^^  where  m  \^  2i  constant  to  be  determined.     ' 

d  d^ 

Since  —e'"''  =  me"""",  — ^'«^  =  m^e*""",  etc. ;   the  result,  after 
ax  dx^ 

rejecting  the  factor  ^'^^j  is 

A^m*'  -\- A^m^--^  -\-  .  ,  .  -\- An-rm  -\- An  =^  o,     .     .     (2) 

an  equation  of  the  n\h.  degree  to  determine  m.  Hence,  if  m  j, 
satisfies  equation  (2),  e*^^  is  an  integral  of  equation  (i) ;  and,  if  . 
m^^m^...  mn  are  n  distinct  roots  of  equation  (2), 

y  =  C.e^x^  +  C^e^^  +   .  .  .   +  CnC'^nX         ...       (3) 

is,  by  Art.  93,  the  complete  integral  of  equation  (i). 
For  example,  let  the  given  equation  be 

d^y        dy 

— —  —  -^  —  2V  =  o : 

dx'        dx  ^  ' 


94  LINEAR  EQUATIONS:   CONSTANT  COEFFICIENTS.  [Art.  95. 

the  equation  to  determine  m  is 

m^  —  m  —  2  =  o, 
whose  roots  are  —  i  and  2  ;  therefore  the  complete  integral  is 

96.  Denoting  the  symbol  —  by  D,  equation  (i)  of  Art.  95 
may  be  written 

{AJ)^  +  A,l>-^  +  .  .  .  +  An-.D  -f-  An)y  =  o, 

or,  symbolically, 

AJ^)y  =  o, .     (I) 

in  which  /  denotes  a  rational  integral  function.  With  this 
notation,  equation  (2)  of  the  preceding  article  becomes 

f{m)  =  o ; 

and,  denoting  its  roots,  as  before,  by  ;«„  7n^  .  ,  .  ntn,  equation 
(i)  may,  in  accordance  with  the  principles  of  commutative  and 
distributive  operations  (Diff.  Calc,  Art.  406  et  seq),  be  written 
in  the  form 

{D  -  m,)  {D  -  7n;)  .  .  .  {D  -  m„)y  =  o.  .     .     .     (2) 

This  form  of  the  equation  shows  that  it  is  satisfied  by  each 
of  the  values  of  j/  which  separately  satisfy  the  equations 

(Z>  —  m,)y  =  0,     (Z)  —  m2)y  =  0,     .  .  .    (Z>  —  m„)y  =  o ; 

that  is  to  say,  by  each  of  the  terms  of  the  complete  integral. 


§  IX.]  CASE   OF  EQUAL   ROOTS.  95 

Thus   the   example  given    in    the   preceding  article   may  be 

written 

(i?+  i)(Z>-  2)7  =  0, 

and    the   separate   terms    of   the   complete   integral   are   the 
integrals  of 

{D  +  \)y  —  o    and    (Z>  —  2)y  =  o, 

which  are  d^"*  and  Cie""^  respectively. 


Case  of  Equal  Roots* 

97.  When  two  or  more  roots  of  the  equation  f{m)  =  o 
are  equal,  the  general  solution,  equation  (3),  Art.  95,  fails  to 
represent  the  complete  integral ;  for,  if  m^  =  nti,  the  corre- 
sponding terms  reduce  to 

in  which  C^  +  C^  is  equivalent  to  a  single  arbitrary  constant. 
It  is  necessary  then  to  obtain  another  particular  integral ; 
namely,  a  particular  integral  of 

{D  -  m,yy  =  0, (i) 

in  addition  to  that  which  also  satisfies  (D  —  m^y  =  o. 
This  integral  is  obviously  the  solution  of 

{D  —  m^y  =  Ae""^"", (2) 

for,  if  we  apply  the  operation  D  —  m^  to  'both  members  of  this 
equation,  we  obtain  equation  (i).  Equation  (2)  is  a  linear 
equation  of  the  first  order,  and  its  complete  integral  is 


=  \Adx  = 


^-m^xy   —    14^^   ^   Ax   -^   By 


96  LINEAR  EQUATIONS:   CONSTANT  COEFFICIENTS.  [Art.  97. 

or 

y  =  e^^'i^Ax  +  B) (3) 

Hence  the  terms  of  the  integral  of  /(B)j/  =  o  corresponding 
to  a  double  root  of  /(m)  =  o  are  found  by  replacing  the 
constant  of  integration  by  Ax  +  B.  For  example,  given 
the  equation 

dx^         dx^       dx 
or 

D{D  -  ^Yy  =  o, 

the  roots  of  fipi)  =  o  are  o,  i,  i,  and  the  complete  integral  is 

y  z=C  -\-  e^{Ax  +  B). 

98.   If  there  be  three  roots  equal  to  m^^  we  have,  in  like 

manner,  to  solve 

{D  -  m,yy  =  o (i) 

But  the  integral  of  this  is  the  same  as  that  of 

{D  —  pt^)y  =  e'^^^iAx  +  B)) (2) 

for,  by  the  preceding  article,  if  the  operation  {D  —  m,Y  be 
applied  to  each  member  of  this  equation,  the  result  will  be 
(D  —  mfy  =  o.     The  integral  of  equation  (2)  is 

e-m,xy  =  U^jc  +  B)dx  =  iAs^  -\-  Bx  -^C; 

t>r,  writing  A  in  place  of  ^A, 

y  =  e^i^iAx'  +  Bx  -hC) (3) 

Hence  the  terms  corresponding  to  a  triple  root  of  /(m)  =  o 
are   found   by  replacing   the   constant  of  integration   by  the 


§  IX.]  CASE    OF'    IMAGINARY  ROOTS.  97 

expression  Ax^  +  Bx  +  C.     In  like  manner,  we  may  show  that 
the  terms  corresponding  to  an  r-fold  root  m,  are 

e'^i^iAx''-^  +  Bx"--^  +  .  .  .  4-  Z). 

In  particular,  if  the  r-fold  root  is  zero,  we  have  for  the  integral 
of 

y  =  Ax"---"  4-  Bx"--^  +  .  .  .  +  Z, 
as  immediately  verified  by  successive  integration. 


Case  of  Imaginary  Roots. 

99.  When  the  equation  f{m)  =  o  has  a  pair  of  imaginary 
roots,  the  corresponding  terms  in  the  complete  integral,  as 
given  by  the  general  expression,  take  an  imaginary  form  ;  but, 
assuming  the  corresponding  constants  of  integration  to  be  also 
imaginary,  the  pair  of  terms  is  readily  reduced  to  a  real  form. 
Thus,  if  w,  =r  a  +  t(3  and  ntz  =  a  —  //?,  the  terms  in  question 
are 

C^^ia  +  zl8  )x   _|.  C^^(a  -  t?  )x  ^   gax(^  C,e^^^   +  C^*?  "  ^^^) .  .       .       (  I ) 

Separating  the  real  and  imaginary  parts  of  e^^"^  and  e-*^^j  the 
expression  becomes 

e^'^liC,  +  C2)cos/3x  +  t{C,  -  C^)  sin  fix}; 

or,  putting  d  -{-  C2  =■  A  and  i(Ci  —  C2)  =  B, 

.      r^{Acosfix  -t  BsmjSx), (2) 

where,  in  order  that  A  and  B  may  be  real,  d  and  C2  in  (i) 
must  be  assumed  imaginary. 


98   LINEAR  EQUATIONS:   CONSTANT  COEFFICIENTS.  [Art.  99. 

As  an  example,  let  the  given  equation  be 

(/?»  +  /^  +  1)7  =  o; 

the  roots  are  —J  ±  \i^l ;   here  a  =  —  J,  ^  =  \^i ;   hence  the 
complete  integral  is 

y  =  e-UAcos^x  +  ^sin^A 

100.  If  the  equation  /(m)  =  o  has  a  pair  of  imaginary 
r-fold  roots,  we  must,  by  Art.  98,  replace  each  of  the  arbitrary 
constants  in  expression  (i)  by  a  polynomial  of  the  (r  —  i)th 
degree ;  whence  it  readily  follows  that  we  must,  in  like  manner, 
replace  the  constants  in  expression  (2)  by  similar  polynomials. 
Thus  the  equation 

or 

{D^  +  ^Yy  =  o, 

in  which  ±/are  double  roots,  has  for  its  integral 

y  =  (A I  +  Bix)  cos  X  +  (A2  H-  B2X)  sin  x. 


The  Linear  Equation  with  Constant  Coefficients  and  Second  Member 
a  Function  of  x. 

Id.  In  accordance  with  the  symbolic  notation,  the  value  of 
y  which  satisfies  the  equation 

AD)y  =  AT    ........     (I) 

is  denoted  by 

^  =  A^/- ^^> 


§  IX.]  THE  INVERSE   OPERATIVE  SYMBOL.  99 

Substituting  this  expression  in  equation  (i),  we  have 

which  may  be  regarded  as  defining  the  inverse  symbol  (2),  so 
that  it  denotes  any  function  of  X  which,  when  operated  upon 
by  the  direct  symbol  f{P),  produces  the  given  function  X. 
Then,  by  Art.  94,  the  complete  integral  of  equation  (i)  is 
the  sum.  of  any  legitimate  value  of  the  inverse  symbol  and  the 
complementary  function  or  complete  integral  of 

AD)y  =  o. 

This  last  function,  which  is  found  by  the  methods  explained  in 
the  preceding  articles,  we  may  call  the  complementary  function 
for  f{D) ;  and  we  see  that  two  legitimate  values  of  the  symbol 

X  may  differ  by  an  arbitrary  multiple  of  any  term  in  the 

complementary  function  ior  f(D) ;  just  as  two  values  of  \Xdx 
or  —X  may  differ  by  an  arbitrary  constant,  which  is  the  com- 
plementary function  for  D. 

102.  With  this  understanding  of  the  indefinite  character  of 
the  inverse  symbols,  it  is  evident  that  an  equation  involving 
such  symbols  is  admissible,  provided  only  it  is  reducible  to  an 
identity  by  performing  the  necessary  direct  operations  upon 
each  member.  It  follows  that  the  inverse  symbols  may  be 
transformed  exactly  as  if  they  represented  algebraic  quantities  ; 
for,  owing  to  the  commutative  and  distributive  character  of  the 
direct  operations,  the  process  of  verifying  the  equation  is 
precisely  the  same  whether  it  be  regarded  as  symbolic  or 
algebraic.     For  example,  to  verify  the  symbolic  identity 

jy  —  a^  2a\D  -a  D  ^  a     ) 


100  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.[Art.  102. 

we  perform  the  operation  D^  —  a'  on  both  members ;  thus 

X  =  ±Ud  +  a)(D  -  a)-l—X-  {D  -  a){D  ^  a^—^x'^ 
2a[_  D  —  a  iJ  -\-  a    J 

=  ±[{D  +  a)X  -  (D  -  a)x'\  =  —2aX  =  X, 
2a[_  J        2a 

the  process  being  equivalent  to  that  of  verifying  the  equation 

_i_  =  -L/_? L_\ 

D'  -  a"       2a\D  -a       D  +  aj 

considered  as  an  algebraic  identity. 

103.  The  symbol X  denotes  the  value  of  y  in  the 

equation  of  the  first  order 


hence,  solving,  we  have 


I 


-^  -  ay  =z  X; 
dx 


X  =  e^Ae-^^Xdx (i) 


D  —  a 
By  repeated  application  of  this  formula,  we  have 

! X  =  — ? — e^Ae-<^^Xdx  =  e^A\e-^^Xdxdx)    (2) 

{D  -  ay  D  -  a      ]  JJ  '    ^  ^ 

and,  in  general, 

Tn^.''  =  ^'\\--Y'''"^'-  •   •  •   (3) 

the  last  expression  involving  an  integral  of  the  rth  order. 


§  IX.]      GENERAL   EXPRESSION  FOR    THE  INTEGRAL.  lOI 


General  Expression  for  the  Integral. 

104.  We  may,  by  means  of  equation  (i)  of  the  preceding 
article,  write  an  expression  for  the  complete  integral  of 
f{P)y  =  X  involving  a  multiple  integral  of  the  7/th  order. 
For,  using  the  notation  of  preceding  articles,  we  may  put 

f{D)  =  (Z>  -  m;){D  ^m,),..{D-  m„); 
whence 

I       Y  _  I I  I  Y 

/(jD)       ^  D  -  m,  D  -  m^"  '  D  -  Mn 


=   ^»«i-^U(w2-/«i)jr       .  .   .      , 


ei^z~^dx\  ,  ,     \e-mnXdx^'j.     .     .     (i) 

but  the  expression  given  below  is  preferable,  involving,  as  it 
does,  multiple  integrals  only  when  the  equation  f(P)  =  o  has 
multiple  roots. 

105.  Let  — — -  be  resolved  into  partial  fractions  ;  supposing 
m^y  m^  .  .  .  m„  to  be  all  different,  the  result  will  be  of  the  form 


/{£>)        D  -  m,       D  -  m^  D  -  ntn 

in  which  N^,  N^ .  .  .  N^  are  determinate  constants  ;   hence,  by 
equation  (i).  Art.  103, 

j^X  =  N,e^Ae-^r^Xdx  +  .  .  .  +  Nne"'nAe-*'^n^Xdx,    (2) 
which  is  the  general   expression  *  for  the  complete   integral 

*  First  published  by  Lobatto,  "Theorie  des  Caracteristiques,"  Amsterdam, 
1837 ;  independently  discovered  by  Boole,  Cambridge  Math,  jfournaly  ist  series. 
vol.  ii.  p.  114. 


102  LINEAR  EQUATIONS:  CONSTANT  COEFFICrENTS.{Ax\..  105. 

when  the  roots  of  /(/>)  =  o  are  all  different ;  each  term,  it  will 
be  noticed,  containing  one  term  of  the  complementary  function. 
When  two  of  the  roots  of  f{D)  =  o  are  equal,  say  ;;/,  =  w^, 
the  corresponding  partial  fractions  in  equation  (i)  must  be 
assumed  in  the  form 

£>  -  m,        {D  -  7n,y' 

and  then  by  equations  (i)  and  (2),  Art.  103,  the  corresponding 
terms  in  equation  (2)  will  be 


\e*"^^  e  -  ""i^Xdx  -\-  N^e""^^    , 


iV;^*"!-^  e - "'I'^Xdx  4-  N^e'"^^ \\e- '^'^''Xdxdx. 

In  like  manner,  a  multiple  root  of  the  r\.\\  order  gives  rise  to 
multiple  integrals  of  the  rth  and  lower  orders. 

106.  When  f{D)  =  o  has  a  pair  of  imaginary  roots,  a  ±  2/?, 
we  may  first  determine,  for  the  corresponding  quadratic  factor, 
a  partial  fraction  of  the  form 


{D  -  ay  +  fi- 


The  corresponding  part  of  the  integral  will  be  found  by  applying 
the  operation  N^D  H-  N^  to  the  value  of 


{D  -  ay  +  ^' 


X. 


Decomposing  the  symbolic  operator  further,  this  expression 
becomes 


2i^\D  -  a  -  /yS        n  -  a  +  ip)      ' 


that  is. 


2/^3  J  2/^  J 


§  IX.]  EXAMPLES.  103 

This  last  expression  is  the  sum  of  two  terms  of  which  the 
second  is  the  same  as  the  first  with  the  sign  of  i  changed ; 
and,  the  first  term  being  a  complex  quantity  of  the  form 
P  -f  iQ  where  P  and  Q  are  real,  the  sum  is  2/*,  or  twice  the 
real  part  of  the  first  term.     Hence 


(Z>  -  a)-  4-/32 
=  the  real  part  of  — (cos /Sjx:  +  /sin/?jc)  U-«^(cosjS^  —  i€m.px)Xdx, 


or 

I 


{D  -  ay  -{-  13^ 

When  a  =  o,  this  result  reduces  to  that  otherwise  found  in 
Arts.  91  and  82. 

Examples  IX. 
Solve  the  following  differential  equations  :  — 

I.     ^  -  5  ^  +  6)^  =  o.  y  =  c,e^^  +  c^e-i^, 

dx^  ax 


2.     6—^  =z  -2  +  y^  y  —  c^e^x  +  c^e-\' 


3.    ^'^(^  ■\-y\=-  (a'  +  ^)^,  >  -  ^/'  +  ^a/'. 


104  I^r^^^EAR  EQUATIONS:  CONSTANT  COEFFICIENTS\Kx\..  I06. 


4.     --^  —  2^  +  SD'  =  o,  _y  =  e'{^Aco%2x  H-  -5 sin  2^). 


y  =  ^,^J^  +  c^e-^  H-  ^sin|(^  -f-  a). 

6.     ^  +  ^-6^  =  0,  ;- =  ....^  +  .,.-3.  +  .,. 

</^3       tf^  ax 


dx^       dx^       dx 


dx^  dx^  dx 


y  =  c^e'^  -\-  e-^{c2  +  ^3^  +  c^x""), 


-  g-^g^«2-«l^-=°' 


_y  =  e^{Ci  +  <r2^)sin;>;  -f-  e^{c^  +  ^4^)cos;t. 


^      dx"  ^ 


,         .  .   x'^xnax   ,    co%ax\oQ,Q.o's>ax 

y  =i  CiCosax  +  CzSmax  H H -2 . 

a  a^ 


§  IX.]  EXAMPLES.  105 


14.      -^  Ar  y  ^  sec^^, 


^  =  ^  cos  (^  +  a)  +  sin:vlog^^t_^HLf  _  i. 


cos^ 


(Z>  -  a)^  +  ^- 


—  (a  cos  /?^  —  *  /?  sin  yS^)  L  -  «-»^  sin  fSxXdx 


1 7.     Show  that ^ J^ 


— I  %max\cos axXdx  —  cosa^  : 

2^1  J  J 


COS  axXdx  —  cos  ax  sin  axXdx 


--i-r 


— ^    cos  ^x    COS  axXdx^  +  sin  «^    sin  axXdx^  . 


y  =  ^i^-2-=p  +  ^^(^jcosjc  +  rjsin:^)  +  LJl\e'2xXdx 

10  J 


—    (3sin;i:  —  cos:v)L?--*cos;i;X/;k: 

—  (3 cos X  4-  sin x)\e-^ sin xXdx  \. 


^ 


15.     ^  +  y  =  tanx,     v  =  ^ cos  (^  +  a)  -  cos ^ log  ^  "^  ^^"'^. 

dx^  ZQSX  (■ 


16.     Show  that ^ ^ 


I06  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  lO/. 


Symbolic  Methods  of  Integration. 

107.  The  foregoing  general  solution  of  linear  equations  with 
constant  coefficients,  Art.  105,  is  theoretically  complete  ;  for  the 
solution  of  a  differential  equation  consists  in  finding  a  relation 
between  x  and  y  involving  only  the  integral  sign.  But,  in  the 
case  of  certain  forms  of  the  function  X  of  frequent  occurrence, 
while  the  evaluation  of  the  integrals  arising  in  the  general 
solution  would  be  tedious,  the  final  result  may  be  very  expe- 
ditiously obtained  by  the  methods  now  to  be  explained. 

In  the  first  place,  suppose  the  second  member  X  to  be  of  the 
form  e^^  \  in  other  words,  let  it  be  required  to  solve  the 
equation 

f{D)y  =  e-- (i) 

Since,  as  in  Art.  95,  D^'e^^  —  aTe^'',  and  f{D)  is  a  sum  of  terms 
of  the  form  Aiy, 

f{D)e-- =  f{a)e-- ; (2) 

whence 

Here  f{a)  is  a  constant ;  and  therefore,  except  when  f{a)  =  o, 
we  may  divide  by  it  and  write 

^"  =  -^^^ (3) 


which  is  the  value  of  y  in  equation  (i).  Thus  we  may,  when 
the  operand  is  of  the  form  Ae^^,  put  D  ^=^  a  'm  the  operating 
symbol  except  when  the  result  would  introduce  an  infinite 
coefficient 


§  X.]  SYMBOLIC  METHODS  OF  INTEGRATION.  10/ 

io8.  In  the  exceptional  case,  equation  (2),  of  course,  still 
holds ;  but  it  reduces  to  f{D)e^^  =  o,  and  thus  only  expresses 
that  e^^  is  a  term  of  the  complementary  function.  In  this  case, 
we  may  still  put  a  for  D  in  all  the  factors  of  f{D)  except 
D  —  a.     Thus,  putting 

we  have 


/{D)  D  -  a  <p{Z>)  <f){a)  D  -  a 

and  hence,  by  equation  (i).  Art.  103, 


(/,«      J 


^    .^«^  _  _L — e'^Ae-'^'^e^'^dx  =   ^^'^^ 


AD)  (/,«       J  ci>{a) 

Again,  if  f{D)  =  {D  ^  ay<l>(D),  so  that  ^  is  a  double  root 
of  f{D)  =  o,  we  shall  have 


■AD)  {D  -  a)^  <i>{D)'^         cf>{a)'    JJ^'^ 


2<f>{a) 


109.    As  an  illustration,  let   it   be   required   to   solve  the 
equation 

g  _;;  =  (..+   I). (I) 

The  complementary  function  is 
The  particular  integral  is 

In  the  first  and   third  terms,  we   may  put   D  =.  2  and  o 


I08  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  IO9. 

respectively,  thus  obtaining  |^'-^  —  i  ;   but  D  =  i  makes  the 
second  term  infinite  ;  hence  we  write 


The  complete  integral  of  equation  (i)  is  therefore 
y  =  e-^^(Acos^x  -f  Bsin^x\  +  e^(C  +  |^)  +  ^e^""  -  i.   ' 

no.  The  value  of  the  particular  integral  in  the  case  of 
failure  of  equation  (3),  Art.  107,  may  also  be  derived  directly 
from  that  equation  by  the  principle  of  continuity.  It  must  be 
remembered  that  properly  the  equation  should  be  understood 
to  contain  the  complementary  function  in  the  second  member. 
Hence,  a  being  a  root  of  /{D)  =  o,  and  at  first  assuming  the 
operand  to  be  e^"^  +  ^^"^^  we  may  write 


Developing  ^^,  the  second  member  becomes 

in  which  the  first  term  is  part  of  the  complementary  function. 
We  may  therefore  write,  for  the  particular  integral, 

i^{a  Jr  h)         \  2  I 

because,  a  being  a  root  of  f{z)  =  o,  f{z)  =z  {z  —  a)(f>{z),  and 
f{a  4-  h)  =  /i<l>{a  -h  h). 


§  X.]  SYMBOLIC  METHODS  OF  INTEGRATION.  IO9 

Now,  making  h  :=  om.  this  result,  we  obtain 

L_  ^ax  —    \ y.gax 

AD)  cl>{a)         ' 

as  before. 

This  is  an  instance  of  a  general  principle  of  which  we  shall 
hereafter  meet  other  applications ;  namely,  that,  when  the  par- 
ticular integral,  as  given  by  a  general  formula,  becomes  infinite, 
it  can  be  developed  into  an  infinite  term  which  merges  into  the 
complementary  function,  and  a  finite  part  which  furnishes  a  new 
particular  integral. 

Again,  when  <^  is  a  double  root,  and  X  =  ^%  the  infinite 
expression  can  be  developed  into  two  infinite  terms  which 
merge  into  the  complementary  function,  together  with  a  finite 
term  which  gives  the  new  particular  integral.  For  example, 
since  h  is  ultimately  to  be  put  equal  to  zero,  we  may  write 


gax  -— ^(a  +  A);r 


{D  ~  ay<i>{D)  <}>{a  +  /i)k' 

gax 


l^\  2  / 


</)(^  +  h)k' 

The  first  two  terms  have  infinite  coefficients  when  h  =  o,  but 
they  belong  to  the  complementary  function  ;  the  third  term  is 
finite,  and  gives  the  particular  integral 

1 e--  =  _f!f!l 

(I?-aycl>(n)  2cl>{a)' 

Case  in  which  X  contains  a  Term  of  the  Form  sin  ax  or  cos  ax, 

III.  We  have,  by  differentiation, 

D  sin  ax  =  a  cos  ax,     D^  sin  ax  =^  —a^  sin  ax, 

lf*'€\Viax—  (^  — a^Y  €v[i  ax  \ 
whence 

f(^iy)^m.ax  —  /{  —  a'')  sin  ax. 


1 10  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  III. 

and,  in  like  manner,  we  obtain 

f{I>)  cos  ax  —  /(—«»)  cos  ax. 
It  follows,  as  in  the  similar  case  of  Art.  107,  that 

and 

_L_cos<.^  =  ^^cos«^, W 

except  when/(— ^*)  =  o.     It  is  obvious  that  we  may  include 
both  these  results  in  the  slightly  more  general  formula 

j^—'SiVi^ax  +  a)  ^       ^       €\xi{ax  +  a). 

For  example,  to  solve 

^  -  >'  =  sin(x  +  a), 

we  have,  for  the  particular  integral. 


^  ^_  ^  sin  {x  -h  a)  =  -J  sin  (a:  +  a), 


Adding  the  complementary  function,  we  have   the  complete 
integral 

y  =  c^e""  +  c^.e-''  —  \€\n{x  +  a). 

112.  In  order  to  employ  equations  (i)  and  (2)  when  the 
inverse  symbol  is  not  a  function  of  D^,  we  reduce  it  to  a 
fractional  form   in  which   the   denominator  is   a  function   of 


§  X.]    SECOND  MEMBER   OF  THE  FORM  ?AXiax   OR   Q,Q<$,ax.    Ill 

Z>^  This  is  readily  done  ;  for  we  may  put  f{D)  in  the  form 
f,{D^)  -f  Df^{D^)y  and  the  product  of  this  hy  f,{D^)  -  Df^{D^) 
will  be  a  function  of  /?^  Moreover,  since  we  have  ultimately 
to  put  D^  =  —a'y  we  may  at  once  put  —a^  in  place  of  D^  in  the 
expression  for  /(/?),  which  thus  becomes 

For  example,  given  the  equation 

{J>  +  D  -  i^y  ^  sin 2^; 
the  particular  integral  is 

I  I        .  D  +  6    . 

sm  2x  =  sm  2x  =  ■ sm  2x 


I>  -^  D  -  2  D-d  ^'-36 

=  _  -^  +  ^  sin  2X  =  -  cos  2-^  +  3  sin  2x ^ 
40  20 

Adding  the  complementary  function 

y  =  C,^^  +  C,.—  -  cos2^  +  3sin2^^ 

20 

113.  The  case  of  failure  of  the  formulae  (i)  and  (2)  of 
Art.  Ill  takes  place  when  the  operand  is  a  term  of  the 
complementary  function.      Thus,  if  the  given  equation  is 

-^  -\-  a^y  z=.  0,0^  ax. 
dx"          ^ 


the  complementary  function  is  A  cos  ax  -\-  B  sin  ax.  Accord- 
ingly, in  the  particular  integral  -—^ — ^  cos  ax,  the  substitution 
D^  =.  —  a^  gives  an  infinite  coefficient.     The  most  convenient 


112  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  1 1 3. 

method  of  evaluating  in  this  case  is  that  illustrated  in  Art.  no. 
Thus,  putting  a  -\-  h  for  a  in  the  operand,  and  developing 
cos  (ax  +  hx)  by  Taylor's  theorem, 

^ cos  (a  +  K)x 

I  /  .  h}x^  \ 

= f  cos  ax  —  sin  ax  ,hx  —  cos  ax . h  •  •  •  |. 

-{a-\-  hy  +  a\  2  I 

Omitting  the  first  term  which  belongs  to  the  complementary 
function,  we  may  write,  for  the  particular  integral, 

I       /      .  hoc^  \ 

cos  (a  4-  h)x  = -( x^max  •\-   —  cos  «x  —  .  .  . ) ; 

2a  -{-  h\  2  / 


D"  +  a 
and,  making  /r  =  o,  we  obtain 


I                        X  sin  ax 
cos  ax  =  , 


D^  +  a^  2a 

and  the  complete  integral  of  equation  (i)  is 


y  =z  A  cosax  +  B  cosax  +  £_!1?L^. 

2a 


Case  in  which  X  contains  Terms  of  the  Form  X^. 

114.  If  an  inverse  symbol  be  developed  into  a  series  pro- 
ceeding by  ascending  powers  of  D,  the  result  of  operating  upon 
a  function  of  x  with  the  transformed  symbol  is,  in  general,  an 
infinite  series  of  functions  ;  but,  when  the  operand  is  of  the 
form  ;r'«,  where  m  is  3.  positive  integer,  the  derivatives  above 
the  mth  vanish,  and  the  result  is  finite.     For  example,  to  solve 

-^  +  2y  =  x^, 
ax 


§  X.]-  SECOND   MEMBER    OF   THE   FORM 'X"^.  II3 

the  particular  integral  is 


Z>  +  2  2  1  +  ID 


=  i(i  -^D  +  \D^  _  iz)3  +  .  .  .);,3 


1 

—  ^ 


(^3  _  3^2  4_  |^_|). 


and  the  complete  integral  is 

J  =  C^-2^  +  \x^  -  \x^  -I-  3^  _  3, 

This  result  is  readily  verified  by  performing  upon  it  the  opera- 
tion D  -\-  2. 

115.  When  the  denominator  of  the  inverse  symbol  is  divis- 
ible by  a  power  of  D,  the  development  will  commence  with  a 
negative  power  of  Z>,  but  no  greater  number  of  terms  will  be 
required  than  would  be  were  the  factor  D  not  present.  For 
example,  if  the  given  equation  is 

(Z)4  +  Z>3  +  D^)y  =  ^3  +  ^x"^ 
the  particular  integral  is 

Since  the  operand  contains  no  power  of  x  higher  than  x"^,  it  is 
unnecessary  to  retain  powers  of  D  higher  than  D^  in  the 
development  of  the  expression  in  brackets.     Hence  we  write 

•>'  =  i-.(^  -  ^  +  ^')(^'  "^  ^^^  =  (i^  ~ 5  ■*"  ^)(^'  ^  3^'^ 

X^     .     X^  X*  ,     ,  ,     ,     , 

= 1 X^  -^  2X^  +  6x, 

20        4         4 


1 14  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  1 1 5. 

in  which  the  last  term  should  be  rejected  as  included  in  the 
complementary  function.     Thus  the  complete  integral  is 

y  = x^  +  xx^  -\-  CiX  -^  C2  +  e-^^ic,  cos^^  +  ^r.sin^^j. 

"^20  \  *         2  2    / 

It  will  be  noticed  that,  had  we  retained  any  higher  powers  of 
D  in  the  final  development,  they  would  have  produced  only 
terms  included  in  the  complementary  function. 


Symbolic  Formulce  of  Reduction, 

116.  The  formulae  of  reduction  explained  in  this  and  the 
following  articles  apply  to  cases  in  which  X  contains  a  factor 
of  a  special  form. 

In  the  first  place,  let  X  be  of  the  form  e^*  V,  V  being  any 
function  of  x.     By  differentiation, 

±e<'xv  =  ^^^  4.  ae^^F, 
dx  dx 

or 

De^^V  =  e^^{D -\-  a)V. (i) 

By  repeated  application  of  this  formula,  we  have 

J>e^^V  =  De^^{D  +  d)V  =  ^-^(Z)  +  ayV\ 

and,  in  general, 

jye^xy  =  ^^(2)  +  ayV, 

Hence,  when  <^(Z>)  is  a  direct  symbol  involving  integral  powers 
of  Dy  we  have 

(l){D)e^^F  =  e^^cjyiD  +  a)K (2) 


§  X.]  SYMBOLIC  FORMULA   OF  REDUCTION.  II5 

To  show  that   this   formula  is  applicable   also  to  inverse 
symbols,  put 

whence 

V  =  ^         Y  • 

and  equation  (2)  becomes 

in  which  Fi  denotes  any  function  of  x^  since  V  was  unrestricted. 
Now,  applying  the  operation  to  both  members,  we  have 


which  is  of  the  same  form  as  equation  (2). 

As  an  example  of  the  application  of  this  formula,  let  the 
given  equation  be 

-^  —   V  =  xe^^. 
The  particular  integral  is 


y  =  ? e'^^x  —  e"*^ ? x 


I  e^"^  I 


X  = 


r  (I  -  iz)  + . .  .)^  =  ^  -  ^^ 
3  39 


1 16  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  1 1/. 

117.  The  formula  of  reduction  of  the  preceding  article  may 
often  be  used  with  advantage  in  the  evaluation  of  an  ordinary 

integral.     For  example,  to  find  Ic"*"^  sin  fixc/x,  we  have,  by  the 

formula, 

-^  e*»^  sin  ?tx  =  e^^  — -^ —  sin  nx  :  '■'■ 

D  D  +  m  ' 

hence 

^'«-*  sin  nxt/x  =  ^'"•^ ~  ^  sin  nx 

J  D^  -  ni" 

{m  —  D)  sin  nx  =  {m  sin  nx  —  n  cos  nx). 


m^  -\-  n^  m^  +  n^ 

It  may  be  noticed  that  equation  (i),  Art.  103,  is  a  case  of 
the  present  formula  of  reduction,  for 

— I — X  =  — l—e^^e-^^X] 
D  -  a  D  -  a 

hence,  applying  the  formula,  we  obtain 

— -^ X  —  e^^—e-^^X  —  e^Ae-^^Xdx\ 

D  -  a  D  J  ' 

in  which  we  pass  from  the  solution  of  a  differential  equation  to 
a  simple  integration.  In  the  above  example,  on  the  other  hand, 
we  employed  the  same  formula  to  reverse  the  process,  the  direct 
solution  of  the  differential  equation  being,  in  that  case,  the 
simpler  process.     Compare  Int.  Calc,  Art.  63. 

118.    Secondly,  let  X  be  of  the  form  xV.      By  successive 
differentiation,  we  have 

DxV   =  xDV    -f  F, 
D^xV  =  xD^V  +  2Z>F, 
D^x  V  =  xD^  V  -f-  3Z>2  V; 
and,  generally, 

lyxV  =  x]>v  -h  riy-^v.   .   ...   .    (i) 


§  X.]  SYMBOLIC  FORMULA   OF  REDUCTION.  11/ 

Now  let  <^{P)  denote  a  rational  integral  function  of  D,  that  is, 
the  sum  of  terms  of  the  form  a^D'' ;  and  let  us  transform  each 
term  of  ^{D)xV  by  means  of  equation  (i).  We  thus  have 
two  sets  of  terms  whose  sums  are  x^ayD'^V  and  ^a^rD^'-'^V 
respectively.  The  first  sum  is  obviously  ;ir0(Z))  F;  and,  since 
ayrD""-^  is  the  derivative  of  a^D^  considered  as  a  function  of  D, 
the  second  sum  constitutes  the  function  <^'{P)  V,     Hence 

(f>(jD)xF  =  x(l>{D)V  -{-  (l)\D)V,    ....     (2) 

where  <^'  is  the  derivative  of  the  function  <^. 

To  show  that  this  formula  is  true  also  for  inverse  symbols, 
put 

whence 


F  =  — ^  V, : 


and  equation  (2)  becomes 


*(^)^^^'  =  ^^'  +  *'(^>^^" 


or 

xF  =  6(D)x 


xF,  =  ct){D)x-^~F,  -  ^i^K, 


in  which  Vj  denotes  any  function  of  x.     Hence,  applying  the 
operation  — — -  to  both  members,  we  have  the  general  formula 

-^-xF=  x~^F-^^^F,.     ...     (3) 

which  is  of  the  same  form  as  equation  (2),  because  —  —  is  the 
derivative  of  the  function  — . 


Il8  LINEAR  EQUATIONS:  CONSTANT  COEFFIC I ENTS\hx\..  1 1 9. 

119.  As  an  example,  take  the  linear  equation 

dy 
ax 

By  the  formula,  the  particular  integral  is 

^sin^=jc— sinjc  —  ^ — —  sinj: 


Z>  -  I  D  -  \  {D  -  1)' 

=  X — — %mx — ^-— 9in^: 

hence 

y  =  — ^.r(cos^  -f-  smx)  —  ^cos^  +  Ce^. 

This  example  is  a  good  illustration  of  the  advantage  of 
the  symbolic  method,  for  the  general  solution  would  give  the 
integral  in  the  very  inconvenient  form 


,y  =  ^^j, 


e-^xsmxdx  +  Ce^ , 

and,  in  fact,  the  best  way  to  evaluate  the  indefinite  integral  in 
this  expression  is  by  the  symbolic  method,  as  in  Art.  117. 

120.  Finally,  let  X  be  of  the  form  x^  V.     Putting  ;ir  F  in  place 
of  V  in  formula  (2),  Art.  118, 

<t>{D)x^V  =  xti>{D)xV  +  <\i'{D)xV) 

and,  reducing  by  the  same  formula  the  expressions  (f>{D)x  V  and 
<f>\D)xVy  this  becomes 

<t>(Z>)x^V  =  x^(j>{D)  V  +  2X(j>\D)  V  +  ^"{D)  V.     .     (4) 

Again,  putting  xV  'for  V  in  this  formula,  and  reducing  as 
before,  we  have 

<l>{D)x^V=  x^<i>{D)V  +  2,x^<i>\D)V+  zx<f>"{D)V+  <\>"\D)V; 

and  by  the  same  process  we  obtain  similar  formulae  for 
r^V,  x^V,  etc.,  the  numerical  coefficients  introduced  being 
obviously  those  of   the  binomial  theorem. 


§  X.]  SYMBOLIC  FORMULA    OF  REDUCTION.  1 19 

As  an  illustration,  let  the  given  equation  be 
By  formula  (4),  the  particular  integral  is 


x'  sm  2x 

D^  -h  I 


2Z>       .  ,     6Z)2  —  2    . 

sm  2;tr  +  2x sin  2^  -\ —  sin  2x 


D^  +  1  (Z)2  +  i)"  (i:)"  +  1)3 

x^  8jc  2  6 

•  —  sin  2;t: cos  2x  -\ sin  2x, 

3  9  27 


and  the  complete  integral  is 


)x^  —  26  .  8jc 


,  .  QJC^   —    20     .  iSX 

y  z=  Ci  cos  ::!£:  +  ^2  sin  jc  —  2 sin  2:x: cos  2X. 

27  9 


Employment  of  the  Exponential  Forms  of  sin  ax  and  cos  ax. 

121.  It  is  often  useful  to  substitute  for  a  factor  of  the  form 
^vaax  or  cos^;r  its  exponential  value,  and  then  to  reduce  the 
result  by  means  of  formula  (2)  of  Art.  116.  For  example,  in 
solving  the  equation 

— ^  -f   y  =  x^  sin  X, 
dx^ 

we  have,  for  the  particular  integral, 

V  = jc^sin^  = ie^^  —  ^-'*-*^): 

but  it  is  rather  more  convenient  to  write,  what  is  easily  seen  to 
be  the  same  thing,  since  e*'^  =  cos;ir  +  z  sin;ir, 

y  =  the  coefficient  of  /  in a:V-*. 


20  LINEAR  EQUATIONS:  CONSTANT  COEFFICIENTS.  [Art.  121. 


Now 

I 


x^e'^  —  e'^-— : x^  =  e'^- 


21   D\  21  4/^  / 

=  (cosAT  +  i^mx^i-—  +  ^  +  — ^; 
\      6         4        4/ 

whence,  taking  the  coefficient  of  /,  and  adding  the  complement- 
ary function, 


Examples  X. 


Solve  the  following  differential  equations  : 

I.     i-2  —   y  =  a:^2^  4-  e^, 
dx^ 


y  =  c,e^  -\-  c^e-^  +  — (3-^  -  4)  +  ~- 


*  This  method  has  an  obvious  advantage  over  that  of  Art.  120  when  a  high 
power  of  X  occurs.  Moreover,  when,  as  in  the  present  example,  the  trigonometrical 
factor  is  a  term  of  the  complementary  function,  it  should  always  be  employed.  For 
it  is  to  be  noticed  that,  in  formula  (3),  Art.  118,  while  two  legitimate  values  of  the 
symbol  in  the  first  member  can  differ  only  by  multiples  of  terms  in  the  comple- 
mentary function  of  ^{D),  two  values  of  the  second  member  may  differ  by  the 
product  of  one  of  these  terms  by  x.  Hence  a  result  obtained  by  the  formula  might 
be  erroneous  with  respect  to  the  coefficient  of  such  a  term.  In  the  example  of 
Art.  119,  the  uncertainty  would  exist  only  with  respect  to  a  term  of  the  form  xex, 
but  it  is  easy  to  see  that  no  such  term  can  occur  in  the  solution.  In  the  example 
of  Art.  120,  a  similar  uncertainty  exists  with  respect  to  terms  of  the  form  jr^sin  jt, 
x^Q.o%x,  x%\xix,  and  x  cosjt,  none  of  which  occur  in  the  solution.  In  the  present 
example,  if  solved  by  the  same  method,  the  uncertainty  would  exist  with  respect  to 
terms  of  the  same  form ;  and,  as  such  terms  do  occur  in  the  solution,  an  error 
might  arise.     See  Messetiger  of  Mathematics,  vol.  xvi.  p.  86. 


§  X.]  EXAMPLES.  121 


dv 

2.  -^  —  2y  =^  x^  +  e^  -\-  cos  2^, 
^^ 

y  =  <r^2j:  _  ^^  _  1(4^3  +  6^2  ^  5^  _|_  2) 

+  J  (sin  2^  —  cos2Jt:). 

3.  — ^  —   2-^  +  _y  =  ^2^3^^^ 
^Jt:^  dx 

y  =  (A  -\-  Bx)e^  +  —-{2x^  -  4-^  +  3)- 
o 

j;  =  {c^x  +  <r2)  sin  jc  +  {c^x  +  c^  cos  x  —  ^x^  sin  a:. 

ax^  ax 

^  +  4JV  =  sm  T,x  -\-  e^  -\r  x^, 
y  =  Acqs  2x  +  ^ sin  2:1;  +  \{e'^  —  sin3::t:)  +  JC^^^  ~  ^)' 

d^y  dy    ,  .  , 

7.  — ^  —  2-^4-  2y  =  <?-^  sm  ;*:  +  cos  ^, 

y  =  <?-^(^cos^  +  ^sin:r)  —  ^jf<?-*^  cos  jr  +  J(cos:v  —  2sin::t:). 

8.  — ^  +   y  =  ;v  sm  2x, 
dx^ 

y  =  A  cos  jx:  4-  -^  sin  jv:  —  ^x  sin  2 jc  —  ^  cos  2;t% 

dy    , 
dx^ 

y  —  A  cos  ^  +  ^  sin  ^  —  —  cos  :t:  +  -  sin  x. 

4  4 

d^v 
10.      — =^  +  4_y  =  2:^3  sm^  ::i!:, 

jK  =  ^  sin  2:*:  ■\-  B  cos  2^  +  ^^    ~  ^^ 

8 

8^3    _    T^x                          AX^   —    -IX^    . 
^-   COS  2X  — ^ —  sm  2X. 

128  64 


122  LINEAR  EQUATIONS:  CONSTANT  COEFFIClENTS.[Axt.  121. 

II.     --^  ^  y  =  ^^COSJP, 

y  =  Complementary  Function  —  ^r-^  cos  x. 

,,.     ^  +  ^  =  sinf^csini;.,  j,  =  92i|?  +  £i«15  +  c.  F. 

tfjr  126  12 

13.     T^  +  32  ^  +  48;^  =  ^<?-^-*',      >-  =  ^^{x^  +  a:0  +  C.  F. 
ax*  ax  144 


^+2^ 
^/a:^  ^;»^ 


y  = h  ^ cos  ^  M-  C.  F. 

12  48 


15.     -^  —  2-/  +  4;;  =  <?^cos;r, 
dx^  ax 

y  -  ^(3sm^  -  cos  a:)  H-  C.  F.     (Compare  Ex.  IX.,  18.) 
20 


[6.     i—-  4-  I  j  >;  =  ^2  ^  x-\ 

y  =  e-^{c^  +  CiX  +  Cjpc^)  -{-  x^  —  6x  +  12  -j-  e- 


dxK 


17.  (D  -\-  b)*'y  =  cos^^, 

4-  {a^  +  ^2)2cos(«jc  —  «cot~'-]. 

18.  Expand  the  integral  \oc"e^dx  by  the  symbolic  method. 

y.x»e^  =  e^lx*'  —  nx»-^  +  n{n  —  i)x»-''  —  ...]+  c. 

19.  Prove  the  following  extension  of  Leibnitz'  theorem  :  — 

<j>{D)uv  =  u  .  (l){D)v  +  J^u  .  </)'(Z>)z;  +  —  •  <^''(^)z/  +  .  .  .  , 

2  ! 

and  show  that  it  includes  the  extended  form  of  integration  by  parts, 
Int.  Calc,  Art.  74. 


§  X.]  EXAMPLES.  123 

20.  In  the  equation  connecting  the  perpendicular  upon  a  tangent 
with  the  radius  of  curvature, 

(Diff.  Calc,  Art.  349),  /  and  <^  may  be  regarded  as  polar  coordinates 
of  the  foot  of  the  perpendicular.  Hence  show  that,  if  the  radius  of 
curvature  be  given  in  the  form  p  =  /(</>),  the  equation  of  the  pedal  is 

r=^cos(0-|-a)+^^^-l— /(^), 

and  interpret  the  complementary  function  (W.  M.  Hicks,  Messenger  of 
Mathematics,  vol.  vi.  p.  95). 

21.  The  radius  of  curvature  of  the  cycloid  being  p  =  4«cos(^, 
find  the  equation  of  the  pedal  at  the  vertex.  r  =  2aQ  sin  Q, 


124  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS\hx\.,  122. 


CHAPTER  VI. 

LINEAR   EQUATIONS   WITH   VARIABLE   COEFFICIENTS. 

XL 

The  Homogeneous  Linear  Equation, 
122.  The  linear  equation 

dx*^  dx»  -  ^ 

in  which  the  coefficient  of  each  derivative  is  the  product  of  a 
constant  and  a  power  of  x  whose  exponent  is  the  index  of  the 
derivative,  is  called  the  homogeneous  linear  equation.  The 
operation  expressed  by  each  term  of  the  first  member  is  such 
that,  when  performed  upon  x"^,  the  result  is  a  multiple  of  x"^ ; 
hence,  if  we  put  y  ■=^  x*^  va  the  first  member,  the  whole  result 
will  be  the  product  ♦of  ;r"'  and  a  constant  factor  involving  m. 
Supposing  then,  in  the  first  place,  that  the  second  member  is 
zero,  the  equation  will  be  satisfied  if  the  value  of  m  be  so  taken 
as  to  make  the  last-mentioned  factor  vanish.  For  example,  if, 
in  the  equation 

X^-^  -{■  2X^  -  2y  ^  o, (i) 

dx^  dx 

we  put  y  =.  X*",  the  result  is 

\m{m  —  i)  -\-  2m  —  2']x^  =  o; 


§  XI.]  THE   HOMOGENEOUS  LINEAR  EQUATION.  1 25 

hence,  if  m  satisfies  the  equation 

m^  -^  m  —  2  —  o, (2) 

x^  is  an  integral  of  the  given  equation.  The  roots  of  equation 
(2)  are  i  and  —2,  giving  two  distinct  integrals;  hence,  by 
Art.  93, 

y  =  CyX  +  C^X-"^ 

is  the  complete  integral  of  equation  (i). 

The  Operative  Symbol  &. 

123.  The  homogeneous  linear  equation  can  be  reduced  to 
the  form  having  constant  coefficients  by  the  transformation 
x  =.  e^.     For,  ii  x  =  e^,  we  have  (Diff.  Calc,  Art.  417) 


ti        d  ^d^        did         \ 

X—  =  — ,  x^—  =  —  (  —  —  I J ; 


dx       de  dx"       dd\de 

and,  in  general, 
d 


dxr 


did        \         Id  ^    \ 


so  that  in  the  transformation  each  term  of  the  first  member  of 
the  given  equation  gives  rise  to  terms  involving  derivatives  with 

respect  to  B  with  constant  coefficients  only.     Denoting  —  by  D, 

uu 

the  equation  is  thus  reduced  to  the  form 

/{D)y  =  o (i) 

in  which/ is  an  algebraic  function  having  constant  coefficients. 


126  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS\Kxl.  1 23. 


Now,  if  we  put  ^  for  the  operative  symbol  x-— ,  the  trans- 

(IX 


forming  equations  become 


dx  do^ 


and,  in  general. 


^£:  =  ^(^-  i)(^-  2)...(^-r+  i); 


and  the  result  of  transformation  is 

f{^)y  =  0,* (2) 

in  which  /  denotes  the  same  function  as  in  equation  (i),  but  x 
is  still  regarded  as  the  independent  variable.  As  an  example 
of  the  transformation  of  an  equation  to  the  form  (2),  equation 
(i)  of  Art.  122  becomes 

[^(^  -  i)  -\-  2&  —  2]y  =  o, 
or 

(&^  +  O  —  2)y  =  o. 

124.  The  operator  &  has  the  same  relation  to  the  function 
X"'  that  D  has  to  e"'^ ;  for  we  have 

^x'"  =  mx»*,     O^x""  =  ;//2^'«,    .  .  .   ^''x**^  =  m^x^  ] 
whence 

f{&)x*^  =  /{m)x'" (i) 

/  /V  * 

*  The  factors  x  and  —  of  the  symbol  x —  are  non-commutative  with  one 
dx  dx 

another,  and  the  entire  symbol,  or  iJ,  is  non-commutative  both  with  x  and  with  D ; 

but  it  is  commutative  with  constant  factors,  and  therefore  is  combined  with  them  in 

accordance  with  the  ordinary  algebraic  laws. 


§  XI.]  THE   OPERATIVE  SYMBOL   &.  12/ 

Thus  the  result  of  putting  j/  =  x""  in  the  homogeneous  linear 
equation 

/W>'  =  o (2) 

is  /{m)x^  =  o  ;  whence 

Am)  =  o (3) 

Accordingly,  it  will  be  noticed  that  the  process  of  finding  the 
function  of  m,  as  illustrated  in  Art.  122,  is  precisely  the  same 
as  that  of  finding  the  function  of  -d-,  as  illustrated  in  Art.  123. 

If,  now,  the  equation  /(m)  =  o  has  n  distinct  roots,  m^j  m^ 
.  .  .  w«,  the  complete  integral  of  f{^)y  =  o  is 

y  =  C^^'^i  +  C:,x'"'^  +  .  .  .  +  C«^'«« ;  ....     (4) 

the  result  being  the  same  as  that  of  substituting  x  for  e"^  in 
equation  (3),  Art.  95. 


Cases  of  Equal  and  Imaginary  Roots. 

125.  The  modifications  of  the  form  of  the  integral,  when 
y(^)  ==  o  has  equal  roots,  or  a  pair  of  imaginary  roots,  may  be 
derived  from  the  corresponding  changes  in  the  case  of  the 
equation  with  constant  coefficients.  Thus,  when  f{d)  =  o  has 
a  double  root  equal  to  m,  we  find,  by  putting  x  in  place  of  ^^, 
and  consequently  log  jir  in  place  of  x,  in  the  results  given  in 
Art.  97,  that  the  corresponding  terms  of  the  integral  are 

x'"(A  -\-  Blogx). 

In  like  manner,  when  a  triple  root  equal  to  m  occurs,  the  cor- 
responding terms  are 

x^lA  +  Blogx  +  C{logxy^, 
and  so  on. 


128  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS,[Ar\..  1 25. 

Again,  when  f{d)  =  o  has  a  pair  of  imaginary  roots,  a  ±  y8/, 
we  infer,  from  Art.  99,  that  the  corresponding  terms  of  the 
integral  may  be  written 


The  Particular  Integral. 

126.  The  homogeneous  linear  equation,  in  which  the  second 
member  is  not  zero,  may  be  reduced  to  the  form 

The  complementary  function,  which  is  the  integral  of  f{d)y  =  o, 
is  found  by  the  method  explained  in  the  preceding  articles. 

The  determination  of  the  particular  integral,  which  is  sym- 
bolically expressed  by  —  X,  may,  by  the  resolution  of  — —  into 

partial  fractions,  be  reduced  to  the  evaluation  of  expression  of 
the  form 

— ^—X,     1 X,    etc. 

&  -  a    '     {&  -  ay 

Compare  Art.  105.     The  first  of  these  expressions  is  the  value 
of  y  in  the  equation 

(^  -  a)y  =  X, 
or 

"I  -  "-^  =  ^' 

a  linear  equation  of  the  first  order,  whose  integral  is 
x-'^y  =  Xx-'^-^Xdx) 


§  XL]  THE  PARTICULAR  INTEGRAL.  1 29 


hence 


— ^—X  =  x'^lx-'^-^Xdx (i) 


Again,  applying  the  operation to  both  members  of  this 

^  —  a 

equation,  and  reducing  by  means  of  the  same  equation, 

1 X  —  — ^ — x^Xx-'^-'-Xdx  =  xAx-Ax-'^-'^Xdx^  \     (2) 

{^  -  ay  ^  -  a     ]  J        J  '     ^  ' 

and,  in  general, 

^— — X  =  x4x-Ax-A  .  .  .  \x-''-^Xdx*'.    .     .     (3) 

i&-a)r  J        J        J         J  ^^^ 

127.  Methods  of  operating  with  inverse  symbols  involving  & 
applicable  to  certain  forms  of  the  operand  X,  and  analogous  to 
those  given  in  the  preceding  section  for  symbols  involving  D, 
might  be  deduced.  The  case  of  most  frequent  occurrence  is 
that  in  which  X  is  of  the  form  x^.  From  equation  (i), 
Art.  124,  it  follows  that,  except  when  /{a)  =  o, 

In  the  exceptional  case,  «  is  a  root  of  /(^)  =  o,  and  f{%)  is  of 
the  form  (i9-  —  aY^{^)  where  <^{a)  does  not  vanish ;  hence 

I  I  _  _i I 

{^  ^  ay'^i^)       ~  i\>{a)  (^  -  ay 

But,  by  formula  (3)  of  the  preceding  article, 


130  LINEAR  EQUATIONS:    VARIABLE  COEFFICIENTS.  [Art.  12/. 

As  an  example,  let  the  given  equation  be 

iif*  -^  4-  2x-^  ^  2y  =  x^  +  x: 
dx*  dx 

the  complementary  function  was  found  in  Art.  122  ;  for  the 
particular  integral,  we  have 


I  -   ,        I  I 

•^  ^   +   ^—2  ^—    I^-t-2 

^3        I        I  _  -^^    I    x[dx 

1 —  — X  —  —  i        —  . 

10       3^—1  10       2>]  ^ 

Hence 

y  =  c^x  -^  c^x-"*  +  A^3  +  l^log^^c. 


Symbolic  Solutions. 

128.    The   first   member  of   any  linear  equation   may   be 
written  symbolically 

AD,  x)y. 

In  the  case  of  the  linear  equation  with  constant  coefficients, 
the  operator  is  a  function  of  D  only.  In  the  case  of  the 
homogeneous  equation  considered  in  the  preceding  articles, 
the  operator  is  capable  of  expression  as  a  function  of  the 
product  xD  which  we  denoted  by  ^.  Examples  occasionally 
occur  in  which  /(Z>,  x)  admits  of  expression  as  a  function  of 
some  other  single  symbol  which,  like  ^,  involves  D  only  in  the 
first  degree.  In  these  cases,  the  equation  is  readily  solved  in 
the  manner  illustrated  below. 
Given  the  equation 

S  _  2bx^  +  b^x^y  =  o. 
dx^  dx 


§  XL]  SYMBOLIC  SOLUTIONS.  13! 

Since  {D  —  bx){Dy  —  bxy)  =  By  —  bxBy  -^  by  —  bxDy  +  b'^xy, 
the  equation  may  be  written  in  the  form 

(Z?  —  bxYy  -ir  by  ^  o\ 
or,  putting  t^iox  D  —  bx, 

{V  +  b)y  =  o, 

in  which  the  operator  is  expressed  as  a  function  of  C    Resolving 
it  into  symbolic  factors,  we  have 

and   the   two   terms   of  the   integral  satisfy  respectively  the 
equations 

{K  —  i^b)y  —  o     and    (C  +  isib)y  =  o. 

The  first  of  these  equations  gives 

{D  —  bx  —  i\Jb)y^  =  o, 
or 

^  =  {bx  +  islb)dx', 

and,  integrating, 

logy,  =  ^bx^  +  lybx  +  ^i, 
or 

yi  =  C,e'^^^^(cosx^b  +  isinx^b). 

In  like  manner,  the  second  equation  gives 

J2  =  C2e'^^^^(cosx)Jb  —  isinx^b). 

Adding,  and  changing  the  constants,  as  in  Art.  99,  we  have, 
for  the  complete  integral, 

y  =  eh^^  {A  cos  x\Jb  4-  Bsinxsjb). 


132  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS\hx\..  128. 

When  the  second  member  of  the  given  equation  is  a 
function  of  Xy  the  particular  integral  is  found  by  resolving 
the  inverse  symbol  into  partial  fractions,  as  in  Art.  126,  the 
evaluation  of  each  term  depending  on  the  solution  of  a  linear 
equation  of  the  first  order. 

129.  The  symbolic  operator  can  sometimes  be  resolved  into 
factors  which  are  of  the  first  degree  with  respect  to  D,  but  are 
not  expressible  in  terms  of  any  single  operating  symbol.  In 
these  cases,  the  factors  are  non-commutative ;  the  equation 
can  still  be  solved,  but  this  circumstance  materially  alters  the 
mode  of  solution.     For  example,  the  equation 

g  -  (^^  +  ^)|  +  (^'  -2x)y  =  X     .     .    .     (I) 

may  be  written  in  the  form 

{D-x){D-x^)y^X',      .....     (2) 
for,  by  differentiation, 

(£  -  ")(l  -  "'•^)  =  S  -  "'I  -  ^^-^  -  "I + "'^- 

The  complementary  function  satisfies  the  equation 

{D-x){D-  x^)y=^o', (3) 

and  it  is  evident  that  the  solution  of 

(Z)  -  x-)y  =  o, 

* 

which  is  jK  =  Ce^'^y  satisfies  equation  (3).  But,  since  we  cannot 
reverse  the  order  of  the  symbolic  factors,  equation  (3)  is  not 
satisfied  by  the  solution  of  {D  —  x)y  =  o. 


§  XL]  NON-COMMUTATIVE  SYMBOLIC  FACTORS.  1 33 

130.  To  solve  equation  (2),  put 

(Z>  -  x^y  =  z;;     .......     (4) 

then  the  equation  becomes 

{D-~x)v^X, (5) 

a  linear  equation  of  the  first  order  for  v.     Solving  equation  (5), 
we  have 

and,  substituting  in  equation  (4),  we  have,  by  integration, 

;;  =  e\Ae-\^'^^lAe-hx-'Xdx'  +  c,e\^^  e-l^^  +  h^^dx  +  c^e\^.,     (6) 

131.  The  solution  of  the  general  linear  equation  of  the  first 

order 

(^D^-  P)y^  X 

may  (see  Art.  34)  be  written  in  the  symbolic  form 
"        X  =  e-l'^Ael'^'^Xdx, 


jD  -{-  P 

which  includes  the  complementary  function  since  the  integral 
sign  implies  an  arbitrary  constant.  In  accordance  with  the 
same  notation,  the  value  of  j,  in  equation  (2),  would  be  written 

^  ~  D  -  x^D  -X    ' 

which  is  at  once  reduced  to  the  expression  (6)  by  the  above 
formula.  It  will  be  noticed  that  the  factors  must,  in  the  in- 
verse symbol,  be  written  in  the  order  inverse  to  that  in  which 
they  occur  in  the  direct  symbol. 


134  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  131. 

In  obtaining  this  solution,  the  non-commutative  character 
of  the  factors  precluded  us  from  a  process  analogous  to  the 
method  of  partial  fractions,  Art.  105  ;  we  have,  in  fact,  only  a 
solution  analogous  to  equation  (i)  of  Art.  104. 

Examples  XI. 
Solve  the  following  differential  equations  :  — 

ax^  dx^  dx  X 

3-    2^"  -|  +  zx-^  -  2>y  =  X, 
dor  dx 

y  =  c,x  +  c,x-i  +  i^[^  -  ix-^{xhXdx. 

4.  x^-^^  —  2^  =  0,  y  =  c,x^  +  ^2  +  ^jlog^. 

5.  x^—^^  +  4x-^-\-2y  =  e^,      y  =  c,x-^  +  ^^^-^  +  x-^e^. 

6.  ;e^  +  sx"^  +  2^  =  X, 

dx^  dx^  dx 

y  =  A  cos  logx  -\-  B  sin  log  ji;  +  C  -f  j^x^. 

>'  =  x{A  +  ^logjc)  +  Cjc-^  +  }^-MogJc. 

8.     ^^  +  6^'^  +  9^^  +3^1'  +  ^=  4;., 
dx*  dx^  dx^  dx 

y  —  (c^  -\-  C2  log  x)  cos  log  X  +  {c^  -^  C4  log  ^c)  sin  log  x  +  x. 


§  XL]  EXAMPLES.  135 

9.     x'^—^  +  2X^-^  4-  2j/  =  \o[x  -\--\ 
dx^  dx^  \         xj 

y  =  ^(^coslog;*:  +  B^mXogx  +  5)  +  x-^{C  +  2logA:). 

10.  x"^—^  +  2x'^—^  —  jt^  +_>'  =  xXogx, 

dx^  dx^  dx 

J.  =  £i  +  ^r..  +  c,\ogx  -  te£)!  +  ('°g^)n. 

^        L  8  12    J 

11.  ^  +  4-^^  +  4-^;'  =  0,       y  =  ce-^""-^^  +  ^^^-^  +W2. 

12.  Prove  that 

d^x'^Xogx  =  r^x^logx  +  nr^-'^x^, 

13.  Prove  that 

both  when  /(^)  is  a  direct,  and  when  it  is  an  inverse,  symbol. 


XII. 

Exact  Linear  Equations. 

132.    Using  accents  to  indicate  the  derivatives  of  y  with 
respect  to  x,  the  linear  equation  of  the  «th  order  is 

/>,/«)    +   p^yin-^)   +    .   .   .    4.   P^_^y"  +   p^_J  +    p^^   =    X        (l) 

To  ascertain  the  condition  under  which  this  represents  an  exact 
differential  equation,  and  to  find  its  integral  when  such  is  the 
case,  we  shall  employ  an  extension  of  the  method  of  Art.  84, 
which  consists  in  successive  subtractions  of  exact  derivatives 
so  chosen  as  to  reduce  the  order  of  the  remainder  at  each  step 


136  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  1 3 2. 

until  we  arrive  at  a  remainder  which  is,  or  is  not,  obviously 
exact.  Since  the  second  member  of  equation  (i)  is  a  function 
of  X  only,  the  equation  is  exact  if,  the  subtractions  being  made 
from  the  first  member,  the  coefficient  of  y  vanishes  in  the 
remainder  of  the  order  zero,  which  contains  no  derivatives  of  y. 
When  this  condition  is  fulfilled,  the  sum  of  the  expressions  whose 

derivatives  have  been  subtracted  will  be  equal  to  \Xdx  +  C. 

The  first  term  of  equation  (i)  shows  that  the  first  of  these 
expressions  is  Poy^*^-^^,  of  which  the  derivative  is 

Subtracting  this  from  the  first  member,  the  remainder  is 

{F,  -  Po')y^"-'^  +  P^y^"-'^  +  . . .  +  l^y; 
or,  putting 

Q,  =  P,-  P,\ 

Q^y{n-i)   +  p^yin-:^)   +    .  .  .    +   P„y. 

In  like  manner,  the  next  expression  whose  derivative  is  to  be 
subtracted  is  Qiy^"~''\  the  next  remainder  being 

and  so  on,  the  values  of  Q2,  Q^y  etc.,  being  • 

Q2  =  P2-  Q/,    Qs  =  P3-  <2/,     etc.      ...     (2) 

The  final  remainder  is  Q„y ;  and  the  condition  of  exactness  is 
that  this  shall  vanish,  that  is  to  say,  Q„  =  o.  If  this  condition 
be  fulfilled,  the  integral  will  be 


Q^yin-i)  +  Q^y(n-2)  +   .  .  .   +  Q„_^y  +  Q^_^y  =    {xt/x  +  C 


(3) 


§  XII.]  EXACT  LINEAR  EQUATIONS.  137 

where  Q,  =  P,,  Q,  =  P,  -  PJ,  Q^  z=  P,  -  P/  +  PJ\  and  in 
general, 

Q.^Pr-  Pr-^    +   Pr'-^   -    .  .  .    ±   Po^""^  l 

and  the  condition  of  direct  integrability  written  at  length  is 

Qn=   Pn-   P./-X    +   Pn'-.   -    .  .  .    ±    Po^"^   =    C    .       .        (4) 

133.  For  example,  to  determine  whether  the  equation 

is  exact,  we  have,  by  the  criterion,  equation  (4), 

^3  =  4  -  14  +  16  -  6  =  o; 

hence  the  equation  is  exact ;  and,  forming  the  successive  values 
of  the  coefficients  Q  by  the  equations  (2),  we  find 

^^'  -  ^>£ + ^5^  -  ^^2 + '^^-^  =  -^  + '" 

which  is  a  first  integral  of  the  given  equation. 

Again,  on  applying  the  criterion  to  this  result,  we  obtain 
4x  —  lox  -{-  6x  =  o ;  hence  it  is  also  exact,  and  its  integral  is 
found,  by  the  same  process,  to  be 


I 


(x^  —  x)-^  +  (2X^  —    i);;  ==  i  +  c,x  +  ^2, 
ax  x 


in  which  a  second  constant  of  integration  is  introduced. 

This  last  result  is  not  exact,  for  2x^  —  i  —  (2>^^  —  i)  is  not 
equal  to  zero  ;   but  it  is  a  linear  equation  of   the  first  order, 


138  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  1 33. 

and  its  solution  gives  for  the  complete  integral  of  the  given 
equation, 

xy>J{x*  —  i)  =  scc-^a:  +  ^V(-^'  —  i) 

+  c^\o%{x  +  v^(^  -  i)]  +  c^ 

134.  The  condition  of  direct  integrability,  equation  (4), 
Art.  132,  contains  the  rth  derivative  only  of  the  coefficient  of 
the  rX\i  derivative  of  y  in  equation  (i) ;  whence  it  is  evident 
that  the  product 

is  an  exact"  derivative  when  s  is  a  positive  integer  less  thaji  r. 
For  example,  x^D^y  is  exact,  because  the^ourth  derivative  of  x^ 
is  zero  ;  its  integral  is 

x^D^y  —  sx^£>y  +  6xDy  —  6. 

When  s  is  negative,  fractional,  or  an  integer  equal  to  or 
greater  than  r,  a  term  of  the  form  xWy,  in  equation  (i),  gives 
rise,  in  equation  (4),  to  a  term  containing  x^-^.  From  this  it 
is  evident  that,  if,  in  the  given  equation,  we  group  together  the 
terms  of  the  specified  form  in  such  a  manner  that  s  —  r  has 
the  same  value  for  all  the  terms  in  a  group,  it  is  necessary,  in 
order  that  the  equation  may  be  exact,  that  each  group  should 
separately  constitute  an  exact  derivative.  If  a  single  group  be 
multiplied  by  x*",  and  equation  (4)  be  then  formed,  we  shall 
have  an  equation  by  which  m  may  be  so  determined  that  the 
group  becomes  exact ;  but,  when  the  given  equation  consists  of 
only  one  group,  it  becomes  a  homogeneous  linear  equation  when 
multiplied  by  x''-',  and  it  is  more  readily  solved  by  the  methods 
already  given  for  such  equations. 

135.  When  an  equation  containing  more  than  one  such 
group  of  terms  is  not  exact,  it  may  happen  that  each  group 


§  XIL]      INTEGRATING  FACTORS  OF  THE  FORM  x"".  1 39 

%. 

becomes  exact  when  multiplied  by  the  same  power  of  x.  For 
example,  the  equation 

2x^{x+i)^^+x{^x  +  z)^-.  zy  =  x.    .    .     (i) 
ax^  ax 

contains  two  groups  of  terms,  in  one  of  which  s  —  r  z=:  i,  and 
in  the  other  s  —  r  =■  o.  Multiplying  by  x^,  and  then  substi- 
tuting in  equation  (4)  of  Art.  132,  we  have 

—  3^^  —  "j^m  +  2)x^  +  ^  —  ^{m  -f-  i)x^ 

-h  2(m  +  $)(m  -h  2)x^  +  ^  +  2{m  -\-  2)(;«  -f-  i)x^  —  o, 

which  reduces  to 

{m  +  2){2m  —  i)x^  +  '^  -\-  {m  +  2)  (2m  —  i)x^  =  o, .     (2) 

the  two  terms  in  this  equation  respectively  arising  from  the 
two  groups  in  equation  (i).  If,  now,  the  value  of  m  can  be 
so  taken  as  to  make  each  coefficient  in  equation  (2)  vanish, 
equation  (i)  becomes  exact  when  multiplied  by  x'''.  In  this 
instance  there  are  two  such  values  of  m  ;  namely,  —2  and  ^. 
Using  the  first  value  of  m,  we  have  the  exact  equation 

a(.  +  i)^{  +  f  7  +  2)1  -  %  =  ^, 
ax^        \         xjdx       x^  x^ 

whose  integral  is 

and,  using  the  second  value,  we  have  the  exact  equation 

2{xi  +  ^t)g  +  (7^^  +  3^^)^  -  3^b  =  x^X, 
whose  integral  is 

2x^(x  4-  i)-i^  —  2xb  =   \x^Xdx  —  C2.     .     .     .      (4) 
ax  J 


140  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS,  [Art.  1 35. 

fe 

Having  thus  two  first  integrals  of  equation  (i),  its  complete 
integral  is  found,  by  elimination  of  y  from  equations  (3)  and 
(4),  to  be 

5(^  +  i)>'  =  c^x  H-  c^x-^  -f  xV^^dx  —  cxr-MjciAT^.  .     (5) 


Symbolical  Treatment  of  Exact  Linear  Equations. 

136.  The  result  of  a  direct  integration  is,  when  regarded 
symbolically,  equivalent  to  the  resolution  of  the  symbolic 
operator  into  factors,  of  which  that  most  remote  from  the 
operand  y  is  the  simple  factor  D.  For  example,  the  two 
successive  direct  integrations  effected  in  Art.  133  show  that 

(^3  _  x)D^  +  (8^2  -  3)Z>2  +  \^xD  -f  4 

=  Z>2[(jt:3  -  x)D  +  2^-1]; 

and,  from  Art.  135,  we  infer  the  two  results, 

2x\x  +  i)D^  +  x{_']x  -t-  3)Z>  -  3 

=  x'D{_2{x  4-  i)i)  H-  5  +  3-^-']  ' 
=  x-^D{_2x^{x  +  \)D  -  2J]. 

137.  If,  in  a  group  of  terms  of  the  kind  considered  in 
Art.  134,  m  be  the  least  value  of  r,  and  q  —  m  be  the  constant 
value  of  i"  —  r,  the  group  may  be  written 

A^(^o  +  A,xD  4-*  A^oc'D^  +  .  .  .)Z>«%     .     .     .     (i) 

where  A^,  A^,  .  .  .  ,  are  constant  coefficients,  and  g  may  be 
negative  or  fractional.  Using  d;  as  in  Art.  123,  to  denote  the 
operator  xD,  the  expression  in  parenthesis  may  be  reduced  to 
the  form  /(^),  and  the  group  to  the  form 

x9/{{^)D^y (2) 


§  XIL]    SYMBOLIC    TREATMENT  OF  EXACT  EQUATIONS.    \\l 

It  is  shown  in  Art.  134  that,  if  m  is  not  zero,  and  q  is  zero  or 
a  positive  integer  less  than  m,  every  term  in  the  expression  (i), 
and  hence  the  whole  expression  (2),  is  an  exact  derivative.  The 
symbolic  transformation  expressing  the  result,  in  this  case,  may 
be  effected  by  means  of  the  formula  deduced  below. 
138.  We  have,  by  differentiation, 

ax    dx  dx^       dx 

or 

D&y  =  &Dy  +  Dy ; 
whence  symbolically 

<yD  =  D{{>  -  i) (i) 

Operating  successively  with  ^  upon  both  members,  we  derive 

^^D  =  &D{&  -  i)    =  D{<>  -  i)% 

&W  =  ^D{&  -  l)^  =  D{d^  -  1)3; 

and,  in  general, 

Now,  since  f{&)  consists  of  terms  of  the  form  A&'^j  it  follows 
that 

/{&)D  =  D/{{>  -  I).* (2) 

*  The  formula  by  which  the  homogeneous  linear  expression  is  reduced  to  the 
form  /(■&)y  is  readily  deduced  from  this  formula.     For  equation  (i)  may  be  written 

xD^y  =  D[&  —  i)y; 
and,  multiplying  by  x, 

x^D^y  =  d{^  -  i)y. 

Changing  the  operand  y  to  Dy,  and  using  equation  (2), 

x^'D^y  =  ^(&  —  i)Dy  =  D(^  —  i)(i?  _  2)y. 

Multiplying  again  by  x, 

x^D^y  -  ^{^  -  !){■&-  2)y', 

and  in  like  manner,  we  prove,  in  general, 

xrDry  =z  ^{^  -  l){^  -  2)  .  .  .  {^  —  r  +  l)y. 


142  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  1 38. 

Again,  operating  with  each  member  of  this  equation  upon 
D  (which  is  equivalent  to  changing  the  operand  from  y  to  Dy)^ 

/{0)D^  =  D/{0  -  i)D  =  D'f{{^  -  2). 

In  like  manner, 

f{0)D^  =  D^f{(^  -  2)D  =  D^/{&  -  3); 

and  in  general, 

/{{y)D'»  =  D-'/{(^  -  m) (3) 

139"  If  ^  is  a  positive  integer  less  than  ;;/,  we  can,  by  this 
formula,  write 

whence 

A^/(^)Z)«'  =  ^{<^  -  i)  ...({>-  ^  +  i)/(0^  -  q)D"'-'!, 

in  which  the  expression  for  the  group  is  reduced  to  the  same 
form  as  when  q  =z  o.  We  may  now  remove  one  or  more  of  the 
factors  of  D"'-^  to  the  extreme  left  of  the  symbol,  thus  effecting 
one  or  more,  up  to  w  —  ^,  direct  integrations,  under  the 
condition  t/iat  711  is  not  zero,  and  that  q  has  one  of  the  values 
O,  I,  2  ...  in  —  I. 

The  equation  giving  the  result  of  m  —  g  integrations  is 

x^/{^)n'"  =  Z>'"-'^{{r  -  m  +  q)  ...{{>  -  m  +  i)/(i'>  -  m). 

140.  In  every  other  case,  the  possibility  of  resolving  the 
operator  into  factors  of  the  required  form  depends  upon  the 
presence  of  a  proper  factor  in  /(i>).  To  show  this,  we  have,  by 
differentiation, 

Dx'J^'y  =  x'f^^Dy  +  (^  4-  i)oc^y', 

whence,  using  Dx^^'  as  a  symbol  of  operation, 

x'^{{^  +  q  +  i)  =  Dx<!^' (i) 


§  XII.]        CONDITIONS   OF  DIRECT  INTEGRABILITY.  I43 

Now,  if  —{q  -\-  \)  is  a  root  of  the  equation  f{d)  =  o,  so 
that  we  can  write 

/W  =  {<^  +  q+  i)(#>W, (2) 

we  shall  have 

x^f{{^)D^  =  Dx<^^'(j){{>)D^ (3) 

We  have  thus  a  second  condition  *  of  direct  integrability,  and 
an  expression  for  the  result  of  integration. 

141.  If  the  first  member  of  a  differential  equation  be 
expressed  in  terms  of  the  form  x^f{d)D''y,  the  conditions 
given  in  Arts.  139  and  140  serve  to  show  at  once  whether 
the  equation  can  be  made  exact  by  multiplication  by  a  power 
of  X.  For  example,  equation  (i)  of  Art.  135,  when  written  in 
the  form  considered,  is 

x\2&  +  ^)Dy  +  (2^  +  3)(^  -  1)7  =  X. 

The  first  term  becomes  exact,  in  accordance  with  the  first 
condition,  when  multiplied  by  x-"" ;  and  the  presence  of  the 
factor  {&  —  i)  shows  that  the  second  term  is  also  made  exact 
by  the  same  factor.  Hence,  by  equation  (3),  Art.  138,  and 
equation  (i),  Art.  140,  the  symbolic  operator  may  be  written 

x^Di{2Q-  +  5)  +  x-\2x^  +  3)]. 

*  This  condition  might  be  made  to  include  that  of  the  preceding  article ;  for 
we  might  first,  by  means  of  equation  (3),  Art.  138,  make  the  transformation 

xgf{^)D^  =  xg-^t  x»tDfnf[-Q  _  m)y 

and  then  the  expression  for  x^D^t,  in  terms  of  1?,  which  is 

^{■^  -  \)  ...{■&-  m  J^  I), 

would,  under  the  previous  condition,  contain  the  factor  i9  +  ^  —  w  +  if,  which,  in 
accordance  with  equation  (i),  should  accompany  x'^~^^.  But,  since  under  no  other 
condition  would  this  happen,  and  since  the  factor  would  not  appear  m/()?  —  m) 
unless  i5  4.  ^  4.  I  had  been  a  factor  of  /(»9),  this  transformation  is  clearly  un- 
necessary. 


144  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  I41. 

Again,  both  terms  of   the  last  factor  fulfil  the  condition  of 
Art.  140  when  multiplied  by  x^y  and  the  expression  becomes 

2x'Dx-\D{x^  ^  x\). 

The  value  of  y  obtained  by  performing  upon  X  the  inverse 
operations  in  the  proper  order  is 


I       f   AXdx 
xl  H-  x^      J  2^" 


in  which  each  integral  sign  implies  an  arbitrary  constant.    The 
expression  is  readily  identified  with  that  given  in  Art.  135. 

It  will  be  noticed  that  whenever  an  equation  becomes  exact 
when  multiplied  by  either  of  two  different  powers  of  x^  it  is 
also  susceptible  of  two  successive  direct  integrations. 


Examples  XII. 
Solve  the  following  differential  equations  :  — 


y  —  c^x  -^  ^{i  —  x')(c2  —  sin-'^),  or 

y  =:   C,X  -h   V^(^    -    1)\C,   -   l0g[^   +   V^C^   -    l)]i. 


§  XII.]  EXAMPLES.  145 

4.     — ^  -f  cos  X  -^  —  2  sin  ^  ^^  —  y  cos  x  =  sin  2x. 
dx^  dx^  dx 


sm^  —  I 


4:2 


%^»  =  ^-pl  +  -p^$  + 


6.     ;cH^  +  2)^-^  +  x{x  +  3)^  -  3^  =  ^, 
^:r^  dx 


(f^)%  =  J(7^i(^.+|f^-)^- 


+  ^i 


/   d'^y    ,         dy   , 
7-     y^-li  +   2;t:-f^  +  3;;  =  ^, 
^2^  dx 


:uii  =  ..M  +  .,J.a|  +  .,. 


8.     (a^  -  x)"^  +  2(2;^:  +  i)^  +  2;;  =  o, 

y  =  ^,(4:^3  _  2^^  -  f:^  -  ^) 

+  x^(x  —  i)\c2  —  4c,  log    ^      |. 

10.     Find  three  independent  first  integrals  of  the  equation  /"  =  X. 
f  =  yC^  +  ^"  ^f  -  /  =  \xXdx  +  r„ 


146  LINEAR  EQUATIONS:  VARIABLE  COEFFICIENTS.  [Art.  I4I. 

11.  Derive  (a)  the  complete  integral  oi  y'"  =  X  from  the  above 
first  integrals,  and  ()3)  the  integral  of  /^  =  A"  in  like  manner. 

(a),    2y  =  oA^dx  -  2x\xXdx  -f  \x^Xdx  +  C.F. 
(y8),    dy  =  xAxdx  -  ixAxXdx  +  3^: Lr^X/jc  -  \x^Xdx  -\-  C.F. 

12.  Solve  the  equation 

dx^  dx 

(a),  as  an  equation  of  the  first  order  for  _/;  (^),  as  an  exact  equation 
when  multiplied  by  a  proper  power  of  x. 

(a),    y=.A+       /^^        +  \{2^x  +  i)-3[(^\^_±^V^.r^.^. 

^'^^^  -^       ^(2VAr  +  I)^     (2v'^,+ i)^J    X' ,^y 

.-/ 1 3y-ir  Show  that  the  .equation 

{2x^  +  6^:^)/^  4-  (13^3  4-  41^^)/" 

+  (11:^2  4-  54^^)/'  —  (ioa:  —  Sx})y'  —  2;;  ="  -X" 

ipay.be  written 

-'  {&  +  iy{2&  Jt^  i){&  -  2)y 
-/^-  =-      \  +  ^i(3^  4.  i)(2i'>  4-  3)(^  +  2)Dy='X, 

and  find  its  integraL 


§  XIIL]      LINEAR  EQUATION  OF   THE  SECOND   ORDER.      I47 

XIII. 

TJie  Linear  Equation  of  the  Second  Order. 

142.  No  general  solution  of  the  linear  differential  equation 
with  variable  coefficients  exists  when  the  order  is  higher  than 
the  first  :  there  are,  however,  some  considerations  relating 
chiefly  to  equations  of  the  second  order  which  enable  us  to  find 
the  integral  in  particular  cases,  and  to  these  we  now  proceed. 

If  a  particular  integral  of  the  equation 

%^-pi^Qy  =  o,.  ....  .   (.) 

in  which  P  and  Q  are  functions  of  x,  be  known,  the  complete 
integral,  not  only  of  this  equation,  but  of  the  more  general 
equation 

can  be  found.  For  let  y^  be  the  known  integral  of  (i),  and 
assume 

y  =  y,v 

in  equation  (2).  Substituting,  we  have,  for  the  determination 
of  the  new  variable  v, 

d^v    ,       dy,  dv     ,     d^y,      1 
dx^  dx  dx  dx^ 


dx  dx 

+    Qy.v  J 


=  X   ....     (3) 


The  coefficient  of  v  in  this  equation  vanishes  by  virtue  of  tHe 
hypothesis  that  7,  satisfies  equation  (i)  ;    thus  the  equation 


148     LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  I42. 

-. — »  .    . — —^ ■  *■  ■  -    .. 

becomes  a  linear  equation  of  the  first  order  for  — •  or  1/.    Hence 

dx 

V  may  be  determined  ;  and  then 

y  ^  y^  Wdx  -f  C,y, 

is  the  integral  of  equation  (2),  the  other  constant  of  integration 
being  involved  in  the  expression  for  v' . 

143.  As  an  illustration,  let  the  given  equation  be 

(I  -  :e)g  +  ^1  -  J-  =  x{^  -  ^)5, 

in  which,  if  the  second  member  were  zero,  y  r=i  x  would 
obviously  be  a  particular  integral.  Hence,  assuming  y  =■  xv, 
and  substituting, 


d' 


—  +  (  2  + )—  ±=  ^(i  —  ^)», 

x^        \         I  —  x^Jdx 

or 

Solving  this  equation,  we  have 

or 

dx  x^ 

and,  integrating, 

V  =  "J(i  -  x^)^  +  ^.[sin-';c  -{-        ~^^    J  +^f 


§  XIII.]  A  PARTICULAR  INTEGRAL  KNOWN.  1 49 

Hence 

y  r=  —\x{i  —  x^)^  +  ^[^sin-^jc  +  (i  —  x^)^^  +  C2X. 

144.  The  simplification  resulting  from  the  substitution 
jj/  =  j/^v  is  due  to  the  manner  in  which  the  constants  enter 
the  value  of  j/  in  the  complete  integral.  For  we  know  that  ^ 
is  of  the  form 

y  ^  ^i>'i  +  ^zyz  +  y, 

where  f^  and  j/2  are  independent  particular  integrals  of  the 
equation  when  the  second  member  is  zero,  and  F  is  a  particular 
integral  when  the  second  member  is  X.    Hence  the  form  of  v  is 

y2     y 

and  that  of  if  is 

-'.fey*©' 

so  that  the  equation  determining  v'  must  be  a  linear  equation  of 
the  first  order.  In  like  manner,  whatever  be  the  degree  of  a 
linear  equation,  if  a  particular  integral  when  the  second  member 
is  zero  be  known,  the  order  of  the  equation  may  be  depressed 
by  unity. 

Expression  for  the  Complete  Integral  in  Terms  of  y^. 

145.  The  general  equation  for  v'y  where  y  in  the  equation 

S+^l  +  e-=^ <'> 

is  put  equal  to  y^v,  and  y^  satisfies 


150    LINEAR  EQUATION  OF   THE  SECOND   ORDER.     [Art.  I45. 


is  [equation  (3),  Art.  142] 
dx 


Solving  this  linear  equation  of  the  first  order,  we  have 

,    \Pdx  ,  [        \Pdx^j       , 

and,  since  j  =  jy^v  =  j,  vdx, 

-\Pdx.         ,  .     -\pdx 

e   J        I       \pdxy,j  ,    .  .  \e   ^ 


y  ^  y 


{'—^iyJ'''^Xdx^  +  c,y,  +  c,y,{'——dx       (3) 


is  the  complete  integral  of  equation  (i)  if  j,  is  an  integral  of 
equation  (2).  Owing  to  the  constants  of  integration  implied 
in  the  integrals,  the  first  term  is,  in  reality,  an  expression  for 
the  complete  integral  :  but  the  last  two  terms  give  a  separate 
expression  for  the  complementary  function  ;  that  is  to  say,  for 
the  complete  integral  of  equation  (2). 

146.  Thus  the  complete  integral  of  equation   (2)  may  be 
written 

y  =  ^ijVi  +  ^2>'2 
where 

\Pdx 

J2  =  ^,1^ —dx (4) 


e   J 


This  expression  may,  in  fact,  represent  any  integral  of  equation 
(2) ;  but,  when  the  simplest  values  of  the  integrals  involved  in 
it  are  taken,  it  gives,  when  y\  is  known,  the  simplest  independent 
integral ;  that  is  to  say,  the  simplest  integral  which  is  not  a 
mere  multiple  of  ^,. 


§  XIII.]      RELATION  BETWEEN  THE    TWO  INTEGRALS.      I5I 

For  example,  in  the  equation 

.d^y  dy    , 

dx^  dx 


assuming,  as  in  Art.  122,  j/  =  x^^  we  have . 

w^  —  4^  +  4  =  o. 

A  case  of  equal  roots  arising,  this  gives  but  one  integral  of  the 
simple  form  y  =  x"^,  namely,  y^  =  x^.      Now,  in  the  given 

equation,  P  =  —-;  hence  ^-j^^-^  =  ;i:3 ;  and,  substituting  in 
equation  (4),  we  have 

^2  =  x^\  —  dx  =  ^c^log:^ 
jx* 

for  the  simplest  independent  integral. 

147.  The  relation  between  the  two  independent  integrals  y, 
and  j/a  may  be  put  in  a  more  symmetrical  form.  For  equation 
(4),  Art.  146,  may  be  written 

^^-; (I) 

whence,  differentiating,  we  obtain 

dy2  dy,         -\pdx  ,  . 

This  is  a  perfectly  general  relation  between  any  two  independent 
particular  integrals  of 

^  +  piv  +  e;-  =  O, 
dx^  dx 


152     LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  1 47. 

but  it  must  be  recollected  that  the  value  of  the  constant  implied 
in  the  second  member  depends  upon  the  form  of  the  particular 
integrals  y^  and  y^.  For  this  reason,  the  relation  is  better 
written 


It  will  be  noticed  that,  in  this  equation,  the  change  of  y^  to  rny^ 
multiplies  A  by  w,  but  the  change  of  y^  to  j^  -I-  my^  does  not 
affect  A. 

148.  We  may  also,  by  introducing  jj,  obtain  a  more 
symmetrical  expression  for  the  particular  integral  of  the 
equation 

than  that  given  in  Art.  145.     For,  since  by  equation  (i),  Art. 

147,  dx  =  d-^,  the  particular  integral  in  equation  (3), 

Art.  145,  may  be  written 

which,  by  integration  by  parts,  becomes 

y=  y\y.e\''''''Xdx  -  y\y,y''''Xdx, 

in  which  \Pdxy  in  the  exponential,   is  to  be  so  taken  as  to 

satisfy  equation  (2)  ;  otherwise,  the  second  member  should 
be  divided  by  the  constant  A  defined  by  equation  (3)  of  the 
preceding  article. 


§  XIII.]    RESOLUTION  OF  THE  OPERATOR  INTO  FACTORS.    1 53 

Resolution  of  the  Operator  into  Factors. 

149.  We  have  seen,  in  Art.  129,  that,  when  the  symbolic 
operator  of  a  linear  equation  whose  second  member  is  zero  is 
resolved  into  factors,  the  factor  nearest  the  operand  y  gives,  at 
once,  an  integral  of  the  equation.  Conversely,  when  an  integral 
is  known,  the  corresponding  factor  may  be  inferred  ;  and,  if  the 
equation  is  of  the  second  order,  the  other  factor  is  found 
without  difficulty. 

For  example,  in  the  equation 

^^  ~  "^^^  -  (9  -  4^);^  4-  (6  -  2>x)y  =  o, 

the  fact  that  the  sum  of  the  coefficients  is  zero  shows  that  e* 
is  an  integral.  The  corresponding  symbolic  factor  is  />  —  i, 
and  accordingly  the  equation  can  be  written 

[(3  -  x)D  -  (6  -  3^)](^  -  ^)y  =  o. 

The  solution  may  now  be  completed  as  in  Art.  130;  thus, 
putting  V  z=z  {D  —  \)y^  we  have 

^  ^  3^  -  6^^^ 

V         X  —  z 

the  integral  of  which  is 

V  =  CeT>^{x  —  3)3. 

Finally,  solving  the  linear  equation 

{D  -  i)y  =  Cei-{x  -  3)3, 

we  have  the  complete  integral 

y  —  Ae^  +  Be^^{^\x^  —  42^^  +  150^  —  183), 

in  which  B  is  put  for  the  constant  \C. 


154    LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  150. 

150.  In  general,  if  y^  denotes  the  known  integral,  and  D  —  y] 
is  the  corresponding  factor, 


(Z?  -  ri)y,  =  0,     or     ^  -  r;;;.  =  o ; 


whence 


.  =  -!-f- .    (I) 

y^  dx 


Now,  in  the  case  of  the  equation  of  the  second  order 

{D^  ^  PD  ^Q)y^o, (2) 

the  other  factor  must  be  Z>  +  P  +  77  in  order  to  make  the  first 
two  terms  of  the  expansion  identical  with  those  of  equation  (2) ; 
thus  we  have 

{D^P  ^'ri){D-'ri)y^o; (3) 

which,  when  expanded,  is 

D^y  +  PDy  -  (^^  + Pr^  +  Ay  =  0.    .     .     .     (4) 

T/ie  Related  Equation  of  the  First  Order. 

151.  If,  regarding  t]  as  an  unknown  function,  we  attempt  to 
determine  it  by  equating  the  coefficients  of  y  in  equations  (2) 
and  (4)  of  the  preceding  article,  the  result  is 

£  +  >y^  +  A  +  <2  =  o (I) 

Hence,  to  any  solution  of  this  equation  of  the  first  order,  there 
corresponds  a  solution  of 

^  +  /'^  +  e>  =  o '    .     .     (.) 

dx^  dx 


§XIII.]      RELATED  EQUATION  OF   THE   FIRST  ORDER.       1 55 

Equation  (i)  is,  in  fact,  merely  the  transformation  of  this  equa- 
tion when  we  put,  as  in  the  preceding  article, 

.    .  .  .  :      '  =  51 <3) 

Although  of  the  first  order,  equation  (i)  is  not  so  simple  as 
equation  (2),  which  has  the  advantage  of  being  linear.  In  fact, 
the  transformation  just  mentioned  is  advantageously  employed 
in  the  solution  of  an  equation  of  the  form  (i).  See  Art.  193. 
Since  the  complete  integral  of  equation  (2)  is  of  the  form 

y  =  c,X,^c,X, (4) 

where  X^^  and  X^  are  functions  of  x,  that  of  equation  (i)  is  of 
the  form 

^        c,X^-\-c,X,        X,  +  cX,'      •     •     •     •     V5; 

which  indicates  the  manner  in  which  the  arbitrary  constant  c 
enters  the  solution. 

The  particular  integrals  of  (i)  produced  by  giving  different 
values  to  c  correspond  to  independent  integrals  of  equation  (2), 
that  is  to  say,  integrals  in  which  the  ratio  c^  :  c^  has  different 
values  ;  the  integrals  in  which  c  =■  o  and  ^  =  00  in  the  expres- 
sion (5)  corresponding  to  the  integrals  X^  and  X^  of  equation  (2). 


The  Transformation  y  =  v/{x). 

152.    If,   in  Art.  142,   we   replace  j,    by  ze/„   an   arbitrary 
function  of  x,  the  result  is  that  the  equation 


S  +  ^^  +  e.  =  x (0 


dx^  dx 


156    LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  1 5 2. 

is  transformed,  by  the  substitution 

y  =  o'.z', (2) 

into 

d^v    I      n  dv  ,    ^  XT  /   \ 

^  +  ^■^+ ^■'' =  ^" <3> 

where 

p.=^^p  +  p, (4) 

w^  dx 

c.  =  -i-!^'  +  ^^  +  a (5) 

a/,   dx^        Wi  dx 
^.  =  - (6) 

Pii  Qiy  and  X^  are  here  known  functions  of  x ;  thus  the  equation 
remains  linear  when  a  transformation  of  the  dependent  variable 
of  the  form  y  =■  vf{x)  is  made. 

153.  The  arbitrary  function  w^  can  be  so  taken  as  to  give  to 
/*!  any  desired  value  ;  thus,  if  /*,  is  a  given  function  of  Xy  we 
have,  from  equation  (4), 

^  =  :^{P,^P)dx', 
whence 

Z£/,   ==   ^  J     ^  J  •  .       .       (7) 

Substituting  in  equations  (5)  and  (6),  we  find,  for  the  values  of 
Ci  and  X^y  in  terms  of  P„ 

^■=e--i(^-'-^)  +  <f -f)  ■••(«) 
and 

\\Pdx 

X.  =  X'- (9) 

h\P^dx 


§  XIIL]  THE    TRANSFORMATION y  —  vf{x).  1 57 

These  equations  may  be  used  in  place  of  equations  (5)  and  (6) 
when  w^  is  given,  P^  being  first  found  by  means  of  equation  (4). 
154.  Equation  (4)  may  be  written 

/*!  =  2— logo/,  -h  P', 
ax 

hence,  when  P  is  a  rational  algebraic  fraction,  if  w^  be  taken  of 
the  form  e'/(^),  where  f{x)  is  a  rational  algebraic  function  of  x, 
P,  will  also  be  a  rational  fraction.  From  this  and  equation  (8) 
it  is  manifest  that,  if  the  coefficients  of  the  given  differential 
equation  are  rational  algebraic  functions,  those  of  the  trans- 
formed equation  will  have  the  same  character  when  w.,  is  of  the 
form  e-f^'^^y  f{x)  being  an  algebraic  function. 
In  particular,  if  the  transformation  is 

• 

y  =  €^'*"vy 

we  have,  since  log  'w^  ==  ax"*", 

P^  —  2max^-^  +  P; 
and  then,  from  equation  (8), 

Q^  =  m^a?x^*^-^  +  max^-^P  -\-  m{m  —  i)(u:'«-2  +  Q. 
If,  for  example,  this  transformation  be  applied  to  the  equation 

f&  ^  ^bx^  -h  b^x'y  =  o, 
dx^  ax 

we  have  P  =  —2bx  and  Q  —  Ifx^ ;  whence 

/\  =  2max^-^  —  2bx, 

Qt  =  m^a^x^^-^  —  2bmax**^  -f-  fn{m  —  i)ax^-^  -\-  d^x^. 


158     LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  1 54. 

If  we  put  m  =  2  and  a  =  ^b,  P,  vanishes,  and  (2i  reduces  to  b ; 
thus  the  transformed  equation  is 

^  +  *-  =  o. 
of  which  the  integral  is 

V  =  Acqs  x^b  +  B€m  x^b. 

Hence  that  of  the  given  equation  is 

y  =  e^^'^ {A  COS,  x^b  H-  ^sin^y/<^), 
agreeing  with  the  solution  otherwise  found  in  Art.  128. 

Removal  of  the  Term  containing  the  First  Derivative, 

155.  If,  in  Art.  153,  we  take  P^  =  o,  the  transformed 
equation  will  not  contain  the  first  derivative.  Distinguishing 
the  corresponding  values  of  w,  Q,  and  X  by  the  suffix  zero, 
equation  (7)  gives 

Wo  =  e     ^        ; (i) 

SO  that  the  transformation  is 

-\\PdX  (        N 

y  =  ve   ^^      , (2) 

and  the  transformed  equation  is 

g  -f-  ^oZ'  =  ^o, (3) 


in  which,  by  equations  (8)  and  (9),  Art.  153, 

a;  =  ^^^W". (5) 


Qo^Q-lP^--^-^ (4) 

2  ax 


§  XIIL]  REDUCTION  TO    THE  NORMAL   FORM,  1 59 

If  the  transformation  y  =  w^v  is  followed  by  the  similar 
transformation  v  =  w^u,  where  w^  and  w^  are  known  functions 
of  ;r,  the  effect  is  the  same  as  that  of  the  single  transformation 
y  =■  WjWzU,  which  is  of  the  same  form.  It  follows  that  the 
equations  which  are  derivable  from  a  given  equation  by 
transformations  of  the  form  y  =  vf{x)  constitute  a  system 
of  equations  transformable,  in  like  manner,  one  into  another. 
Among  these  equations  there  is  a  single  equation  of  the  form 
(3)  which  may  thus  be  taken  to  represent  the  whole  system. 
Accordingly  equation  (8),  Art.  153,  shows  that  the  expression 
for  2o)  in  equation  (4),  has  an  invariable  value  for  all  the 
equations  of  the  system.  The  expression  is  therefore  said  to 
be  an  invariant  for  the  transformation  y  =  vf{x). 

156.  One  of  the  advantages  of  reducing  an  equation  to  the 
form  (3),  which  may  be  called  the  normal  form,  is  that,  if  any 
one  of  the  equations  of  the  system  belongs  to  either  of  the 
classes  for  which  we  have  general  solutions,  the  equation  in 
the  normal  form  belongs  to  that  class.  For,  in  the  first  place, 
if,  in  any  equation  of  the  system,  P  and  Q  have  constant  values, 
equation  (4)  of  the  preceding  article  shows  that  Q^  will  also  be 
constant.  In  the  second  place,  if  any  one  of  the  equations  of 
the  system  is  of  the  homogeneous  form 


^  +  fi  4-  +  ^     y 

dx^        X  dx       x^ 

A  B 

putting  P  =  — ,  and  Q  =  —  in  equation  (4),  we  obtain' 


Q^  -  4B  -  A^  +  2  A 

4x^ 


hence  the  transformed  equation  is  of  the  homogeneous  form. 


l6o    LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  1 5 7. 

157.  As  an  example  of  reduction  to  the  normal  form,  let  us 
take  the  equation 

^  -  2  tan :«:  ^  -  (d7»  -h  i)>'  =  o. 
dx^  dx 

Here   P  =  —  2  tan  x  ;    therefore,  by  equations    (i)  and    (4), 
Art.  155, 

I  tan  X  dx 

«/o  =  <?■'  =  sec.r, 

and 

Q^  -=  _(^2  -^.  i)  _-  xzxi^  X  -}-  %tc^x  =  — tf*. 

Thus  the  transformed  equation  is 

*^  ^  a^v  tss.  o. 
dx" 

The  integral  of  this  is 

hence  that  of  the  given  equation  is 

y  =  ^tcx{c^e'*'^  -f  ^2^-«^). 


Change  of  tht  Independent  Variable. 

158.    If  the  independent  variable  be  changed  from  x\.o  Zy  z 
being  a  known  function  of  x,  the  formula  of  transformation  are 

dy  _  dy  dz 
dx        dz  dx ' 
and 


d^y  _  djyfdzV   ,    dy  d^ 
dx^       dz^  \dxl        dz  dx 


z 
do^' 


§  XIII.]       CHANGE   OF   THE  INDEPEiXDEXT   VARIABLE.       l6l 

Making  these  substitutions,  the  equation 

g^^|^e,  =  x. (0 

is  transformed  into 

\dxl    dz-  ^  V^^  ^^/^^ 

which    is    still    linear,   the   coefficients    being    expressible   as 
functions  of  s. 

159.    If  it  be  possible  to  reduce  a  given  equation  by  this 
transformation   to   the  form   with  constant   coefficients,   it   is 

evident,  from  equation  (2),  that  we  must  have  (  —  )    equal  to 

\dx/ 

the  product   of   Q  by  a  constant.      For  example,  given   the 

equation 

(i  —  x^) — ^  —  x-^  -\-  ni^y  —  o, 
dx^  dx 

in  which  Q  =■ ;  if  transformation  to  the  required  form 

I  —  x^ 

rlv  T 

be  possible,  it  will  be  the  result  of  putting  — 


I 
I 


dx  sjii   -  X')  ' 

whence  2  =  sin-':r.     Making  the  transformation,  we  obtain 


-Z  -^  my  =  o, 

dz' 


which  is  of  the  desired  form.     Its  integral  is 

y  =  A  cos  mz  +  B  sin  mz; 
hence  that  of  the  given  equation  is 

y  ss  Acqs  m  sin- ^x  +  B sin  m  sin- ^x. 


1 62     LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  1 5 9. 

In  like  manner,  if  it  be  possible  to  reduce  the  equation  to 
the  homogeneous  linear  form,  we  must  have  (-^)   equal  to 

the  product  of  Qz^  by  a  constant.  But  this  transformation 
succeeds  only  in  the  cases  in  which  that  considered  above  also 
succeeds  ;  for  it  gives  to  log  z  the  same  value  which  the 
preceding  one  gives  to  z ;  accordingly  it  is  equivalent  to  the 
latter  transformation  followed  by  the  transformation  z  =  log  ^, 
which  is  that  by  which  we  pass  from  the  form  with  constant 
coefficients  to  the  homogeneous  form  (see  Art.  123). 

160.  We  may,  if  we  choose,  so  take  z  as  to  remove  the  term 
containing  the  first  derivative.  Equating  to  zero  the  coefficient 
of  this  term  in  equation  (2),  Art.  158,  we  find,  for  the  required 
value  of  Zy 

z  =  Y~^     ^dx. 

Using  this  relation  to  express  ;»;  as  a  function  of  -s',  the 
transformed  equation  is 


dz^  \dzl^  \dz) 


Examples  XIII. 
Solve  the  following  differential  equations  :  — 


y  =  c,e^  -h  c^{xi  +  3^^  -\-  6x  +  6), 


''  £-"^l+"'(^-^>  =  ^' 


y  z=.  c,x  -\-  C2x\e^~  +  I. 


XIIL]  EXAMPLES.  163 


3-     S-|  +  (-')-  =  ^' 


2.eAe^    ^^dx  •\-  eAe"^    ^^\e    2  "*" 


y  =  c^e^  +  <r2^^^2        ^:x:  +  d'-* 


4.     — =^  —  ax-f-  +  «2(:v  —  i)v  =  o, 
dx^  dx 


ax'' 


5.     (^  +  x")^^  —  2x-^  4-  2);  =  o,      >;  =  c^{xi^  —  a)  +  c^x. 
dx^  dx 


,      d^y  dH        dy   , 

6.     — ^  —  ^— ^ -^  xy  =  o, 

dx^  dx'       dx        ^ 


y  =  c^e^  +  ^2^--^  4-  r. 


/zjc^        A'  ^^t:        x^ 


JeAe^^^-^dx  —  ^-•rU^-*''  +  ^^^Y 


j;  =  c^x-'e*"^  +  f2j«:-2^-»«^. 


^2^J^'     .     ^^  _    2^^  4-    2V  -   o 


^  =  r.jr^  +  c 


,x  +  (r3(::\;Mji;-3^-^^A:  —  xXx-^e-'^dxy 


9.     (2^3  _  «)^  —  6x^-^  +  6^y  =  o, 

y  =  <r,(A:3  +  ^)  +  c^x, 
d'v  dv        (x^  4-  iV  -^ 

H.     x'-^  +  {x  —  ¥(^)j-  +  (i  -  2a:  +  4Jc2)^  ^  o^ 

^  =  ^2jr^^^coslog^^  4-  ^2  sin  log  :r)- 


164    LINEAR  EQUATION  OF  THE  SECOND   ORDER.     [Art.  160. 

12.  x^^  -  inx^  -f-  (w^  +  //  -f  a\x^)y  =  o, 

dx^  lix 

y  =  x"{c^Q.osax  4-  Ci^xwax), 

13.  -^  +  tanjc-^  +  yco^'^x  =  0,  _)'  =  ^,  sin  (sin  a:  +  r,). 

14.  (^^  +  A-^)^-^'  +  A'^  -  m^y  =  o, 

y  =  r,[A  +  V(«'   +  ^^)Y   +  ^^[a-   -   v/(«^   +  A^)]'«. 

15.  ^^^  -    2(A^    4-    A-)^   4-   {X^    +•    2A   +    2)J   =    O, 

dx^  dx 

y  =  e-r{c,x^  +  GA). 

,       d^y    ,    2  dy    ^      ,  ^  sinwA    .    ^  cos //a 

^A^        A-  dx  X  X 

,  txd^y  dy    ,       a^y 

17.        (I    -    X')-/     -    2X^   + ^—   =   O, 

dx^  dx        I  —  x^ 

y:=.c,  COS-  log ^^--^  +  ^.  sin  -  log  ^-^. 

^A^  A  ^/a  x''  X  X 

ig.     ^*  4-  (tan A  —  1)2  ^  —  ;?(;?  —  Ojsec-^A  =  o, 
dx^  dx 

y  =  C^e'^^  -- 1)  tan  X  _^  c^e  -  "  '^"  ■*, 

20.  (a^  4-  ^0' r^  +  2x(a'  4-  A^^)^'  4-  ^^;'  =  o, 

dx^  dx 

y  =   il^ +    £2 . 

^        yj{a^  4-  A^)        \l{a^  4-  ^) 

21.  Derive  equation  (3),  Art.  147,  in  the  form 

y^^  -  y^  =  Ae-^''\     from  <>'   +  P'^-f  +  Qy  =  O, 
dx  dx  dx^  dx 

by  eliminating  Q  and  nitegrating  the  result. 


§  XIII.]  EXAMPLES.  165 

22.     Find   the  symbolic  resolution  of  D^   corresponding  to   the 
integral  x  of  the  equation  D^y  —  o. 


^=(^^9(^-9- 


23.  Find  the  symbolic  resolution  oi  D^  —  i  corresponding  to  the 
integrals  cosh x  and  sinh x  of  the  equation  (Z>^  —  \)y  =  o. 

D^  —  I  =  (jD  +  tanh  x)  (D  —  tanh  x) 

=  (Z>  +  coth:v)(Z>  -  coth:^). 

24.  Show  that  the  ratio  s  of  two  independent  integrals  of 

dx^  ax 

satisfies  the  differential  equation  of  the  third  order 

where  Qo  is  the  function  defined  in  Art.  155. 

25.  Show  that,  if  P  be  expressed  in  terms  of  0,  the  equation  of 
Art.  160  may  be  written 


H'2 


+  Qy  =  X. 


26.     Prove  that,  in  the  equation 

£Z  +  P|' +  e;- =  o, 
dx^  dx 

the  function 

is  an  invariant  with  respect  to  the  transformation  z  =  ^{pc)^ 


1 66  SOLUTIONS  IN  SERIES.  [Art.  1 6 1. 


CHAPTER  VII. 

SOLUTIONS   IN   SERIES. 

XIV. 

Development  of  the  Integral  of  a  Differential  Equation  in  Series, 

i6i.  In  many  cases,  the  only  solution  of  a  given  differential 
equation  obtainable  is  in  the  form  of  a  development  of  the 
dependent  variable  y,  in  the  form  of  an  infinite  series  involving 
powers  of  the  independent  variable  x.  Moreover,  such  a 
development  may  be  desired,  even  when  the  relation  between 
X  and  y  is  otherwise  expressible.  If  we  assume  the  series  to 
proceed  by  integral  powers  of  x,  an  obvious  method  by  which 
successive  terms  could  generally  be  found  is  as  follows.  Sup- 
posing the  equation  to  be  of  the  ;?th  order,  and  assuming,  for 
the  11  arbitrary  constants,  the  initial  values  corresponding  to 
X  =  o  oi  y  and  its  derivatives,  up  to  and  inclusive  of  the 
{n  —   i)th,  the  differential  equation  serves  to  determine  the 

value  of  -^  when  x  =z  o.     Differentiating  the  given  equation, 
ax** 

we  have  an  equation  containing  =^,  which,  in  like  manner, 

dx"  + ' 

serves  to  determine  its  value  when  ;ir  =  o,  and  so  on.     Thus, 

writing   out   the  value   of  y  in  accordance  with    Maclaurin's 

theorem,  we  have  the  values  of  the  successive  coefficients  in 

terms  of  n  arbitrary  constants. 


§  XIV.]  LINEAR  EQUATIONS.  167 

162.  It  would  usually  be  impossible  to  obtain,  in  the  manner 
described  above,  the  general  term  of  the  series.  We  shall 
therefore  consider  only  the  case  of  the  linear  equation  (and 
such  as  can  be  reduced  to  a  linear  form),  in  which  case  we  have 
a  method,  now  to  be  explained,  which  allows  us  to  assume  the 
series  in  a  more  general  form,  and,  at  the  same  time,  enables 
us  to  find  the  law  of  formation  of  the  successive  coefficients. 

Since  we  know  the  form  of  the  complete  integral  of  a  linear 
equation  to  be 

y  =  c,y,  +  ^2^2  +  .  .  .  +  Cnyn  +  y, 

our  problem  now  is  the  more  definite  one  of  developing  in 
series  the  independent  integrals  j^„  jz  •  •  •  JF«,  of  the  equation 
when  the  second  member  is  zero,  and  the  particular  integral  Y 
of  the  equation  when  the  second  member  is  a  function  of  x. 
No  arbitrary  constants,  it  will  be  noticed,  will  now  occur  in  the 
coefficients  of  the  required  series,  except  the  single  arbitrary 
constant  factor  in  the  case  of  each  independent  integral. 


Development  of  the  Independent  Integi-als  of  a  Linear  Equation  whose 
Second  Member  is  Zero, 

163.  We  have  seen,  in  Art.  122,  that  if,  in  the  first  member 
of  a  homogeneous  linear  equation  whose  second  member  is  zero, 
we  put  y  =  Ax^\  the  result  is  an  expression  containing  a  single 
power  of  ;ir ;  so  that,  by  putting  the  coefficient  of  this  power 
equal  to  zero,  we  have  an  equation  for  determining  m  in  such  a 
manner  that  y  =  Ax'*'  satisfies  the  differential  equation,  A 
being  an  arbitrary  constant. 

If  we  make  the  same  substitution  in  any  linear  equation 
whose  coefficients  are  rational  algebraic  functions  of  Xy  the 
result  will  contain  several  powers  of  x.  Let  us,  for  the 
present,  suppose  that  it  contains  two  powers  of  x^  and  also 


1 68  SOLUTIONS  IN  SERIES.  [Art.  1 63. 

that  the  differential  equation  is  of  the  second  order.  The  term 
containing  — ^  in  the  differential  equation  will  produce  at  least 

one  term,  in  the  result  of  substitution,  involving  m  in  the 
second  degree ;  hence  at  least  one  of  the  coefficients  of 
the  two  powers  of  x  will  be  of  the  second  degree  in  m.  Let 
X"*'  and  ;ir'«'-^^  where  s  may  have  any  value,  positive  or  negative, 
be  the  two  powers  of  ;r,  and  let  the  coefficient  of  x*"'  be  of  the 
second  degree.  Now  let  m  be  so  determined  that  the  coefficient 
of  X'"'  shall  vanish,  and  suppose  the  quadratic  equation  for  this 
purpose  to  have  real  roots.  Selecting  either  of  the  two  values 
of  nit  the  coefficient  of  x^'+^  will,  of  course,  not  in  general 
vanish. 

Suppose,  now,  that  we  put  for  y,  in  the  first  member  of  the 
differential  equation,  the  expression  AoX^"  +  A^x*"-*-^,  the  result 
will  contain,  in  addition  to  the  previous  result,  a  new  binomial 
containing  ^„  and  involving  the  powers  ;ir''''+^and  x*"'+''^;  the 
entire  coefficient  of  x""'-^^  will  now  contain  A^  and  A,,  and  may 
be  made  to  vanish  by  properly  determining  the  ratio  of  the 
assumed  constants  A^  and  A^.  In  like  manner,  if  we  assume 
for  J/  the  infinite  series 

y  =  AoX^^  +  AiX'*'  +  ^  +  ^a^'^H-"  H-  . . . , 
or 

y  =  t^ArX^'  +  ^^, 

we  can  successively  cause  the  coefficients  in  the  result  of 
substitution  to  vanish  by  properly  determining  the  ratio  of 
consecutive  coefficients  in  the  assumed  series.  If  the  series 
thus  obtained  is  convergent,  it  defines  an  integral  of  the  given 
equation  ;  and,  since  in  the  case  supposed  there  were  two 
values  of  jn  determined,  we  have,  in  general,  two  integrals. 
If  s  be  positive,  the  series  will  proceed  by  ascending  powers, 
and,  if  s  be  negative,  by  descending  powers,  of  x. 


§  XIV.]      DETERMINATION  OF  THE   COEFFICIENTS.  1 69 

\    164.  For  example,  let  the  given  equation  be 

^  -  "i  -  ^-^ "  ° <■> 

The  result  of  putting  A^x^  for  y  in  the  first  member  is 

m{in  —  1)^0^"^ -2  —  (m  -{-  p)AoX'^ (2) 

The  first  term,  which  is  of  the  second  degree  with  respect  to 
M,  will  vanish  if  we  put 

m{m  —  i)Ao  =  o (3) 

The  exponent  of  x  in  this  term,  or  m',  is  m  --  2,  and  the  other 
exponent,  or  in'  +  s,  is  m ;  whence  s  =  2.  We  therefore  assume 
the  ascending  series 

y  =  2o^r^^'  +  2'-, 

and,  substituting  in  equation  (i),  we  have 

2^5  (w  +  2r){m  -\-  2r  —  i)ArX"'  +  ^''-'' 

—  {in  +  2r  -\-  p)ArX'^  +  ^^\  =  o,     (4) 

in  which  r  has  all  integral  values  from  o  to  co. 

In  this  equation,  the  coefficient  of  each  power  of  x  must 
vanish  ;  hence,  equating  to  zero,  the  coefficient  of  x^  +  ^^-^y  we 
have 

{jH  +  2r){i7t  +  2r  —  i)Ar  —  {m  -\-  2r  —  2  -{-  p)Ar^i  =  o.     (5) 

When  r  =:  o,  this  reduces  to  equation  (3)  and  gives 

m  —  o     or     m  =  I ; 

and  when  r  >  o,  it  may  be  written 

{m  4-  2r){m  +  2r  —  i) 

which  expresses  the  relation  between  any  two  consecutive 
coefficients. 


I/O  SOLUTIONS  IN  SERIES.  [Art.  1 64. 

When  m  =-0y  this  relation  becomes 

_  /  +  2r-  2  .       . 
2r(2r  —  i) 

whence,  giving  to  r  the  successive  values  i,  2,  3  .  .  .,  we  have 

^^  -      3.4    ^^  -  4  !        ^°' 

.    _  /  +  4  .    _   /(/+  2)(/  +  4)    . 


The  resulting  value  of  y  is 

y  =  AJi  +  /^H-/(/  +  2)^ 
L  2  !  4  ! 

+  /(/+   2)(/  +  4)f^  +  ...].      (7) 

Again,  giving  to  m  its  other  value  i,  the  relation  (6)  between 
consecutive  coefficients  becomes 

p  +   2i 

{2r  +  i)2r 


jj         /  +  2r  —  T  D 


whence 


/  +  3  „  (/+!)(/  +  3) 

Tr^^  = s 


Bz  =  — 7-;:—  Bi  =  —^ Bo 


^3  =  -^-  ^2  =   ^-j Bo; 

and  the  resulting  value  of  j/  is 

^  =  5„  1^*  +  (/  +  i)|i  +  0>  +  i)(/  +  3)|!  + . ,  .J.    (8) 


§  XIV.]  CONVERGENCY  OF  THE  SERIES.  17I 

%  . 

Denoting  the  series  in  equations  (7)  and  (8),  both  of  which  are 
converging  for  all  values  of  Xy  by  y^  and  /a,  the  complete 
integral  of  equation  (i)  is 

y  =  Aoy^  +  Boyz (9) 

165.  It  will  be  noticed  that  the  rule  which  requires  us  to 
take,  for  the  determination  of  m,  that  term  of  the  expression 
(2)  which  is  of  the  second  degree  in  m  was  necessary  to  enable 
us  to  obtain  two  independent  integrals.  But  there  is  a  more 
important  reason  for  the  rule  ;  for,  if  we*  disregard  it,  we  obtain 
a  divergent  series.  For  example,  in  the  present  instance,  if  we 
employ  the  other  term  of  expression  (2),  Art.  164,  thus  obtaining 

m  =  —p    and     i-  =  —  2, 

the  resulting  series  is  '     , 

,     /(/+   !)(/  +   2)(/+   3)         ,  1 

2.4  J 

The  ratio  of  the  {r  +  i)th  to  the  rth  term  is 

_  {p+  2r-  2){p+  2r-  I)         _ 


X- 
2r 


and  this  expression  increases  without  limit  as  r  increases, 
whatever  be  the  value  of  x.  Hence  the  series  ultimately 
diverges  for  all  values  of  x. 

When  both  terms  in  the  expression  corresponding  to  (2)  are 
of  the  second  degree  in  m,  we  can  obtain  two  series  in  descend- 
ing powers  of  x  as  well  as  two  in  ascending  powers  ;  and,  in 
such  cases,  the  descending  series  will  be  convergent  for  values 
of  X  greater  than  unity,  and  the  ascending  series  will  be  con- 
vergent for  values  less  than  unity. 


lyi  SOLUTIONS  IN  SERIES.  [Art.  1 66. 


The  Particular  Integral. 

i66.  When  the  second  member  of  a  linear  equation  is  a 
power  of  Xy  the  method  explained  in  the  preceding  articles 
serves  to  determine  the  complementary  function,  and  the 
particular  integral  may  be  found  by  a  similar  process.  Thus, 
if  the  equation  is 

S  -  "I  -  y*-^  =  ^*' 

the  complementary  function  is  the  value  of  y  found  in  Art.  164, 
To  obtain  the  particular  integral,  we  assume  for  y  the  same 
form  of  series  as  before,  and  the  result  of  substitution  is  the 
same  as  equation  (4),  Art.  164,  except  that  the  second  member 
is  x^  instead  of  zero.  Equation  (5)  thus  remains  unaltered, 
while,  in  place  of  equation  (3),  we  have 

m{m  —  i)AoX»^-^  =  x^. 

This  equation  requires  us  to  put 

w  —  2  =  i,     and     m(m  —  i)Ao  =  i  j 
whence 

///  =  S,     and     Ao  =  r%. 

The  relation  (6)  between  consecutive  coefficients  now  becomes 


Ar  = 


/  +  2r  4-  ^ 


(2r  +  |)(2;'4-|)     '-" 
or 

.    _     2(2/  4-  4r  +  i)     yf 

hence 

2(2/ +  5) 
'*•  ~     7.9    ^°' 

_  2(2/  +  9)   ,    _   2'(2;>  +  5)(2/  +  9) 

'^'  -    II.I3     ■  ~       7:9^11:^3        °' 


§  XIV.]      BINOMIAL   AND  POLYNOMIAL   EQUATIONS.  1/3 


and  the  particular  integral  is 


i--i.j,t'"^-^"^+''"^-"""^*'"«-+. 

^         7-9  7-9-II-I3 


If  the  second  member  contained  two  or  more  terms,  each 
of  them  would  give  rise  to  a  series,  and  the  sum  of  these  series 
would  constitute  the  particular  integral. 

Binomial  and  Polynomial  Equations. 

167.  If  we  group  together  the  terms  of  a  linear  equation 
whose  coefficients  are  rational  algebraic  functions  of  x  in  the 
manner  explained  in  Art.  134,  we  can,  by  multiplying  by  a  power 

of  Xy  and  employing  the  notation  ;ir— -  =  ^,  put  the  equation  in 

ax 

the  form 

/xWj^'  +  ^VaWj +  ^V3W;'  + ...  =  o,      .    .     (i) 

in  which  s^,  s^,  .  .  .  are  all  positive,  or,  if  we  choose,  all  negative. 
The  result  of  putting  A^x"*'  for  y  in  the  first  member  is 

AoMm)x»'  +  AJ:,{m)x**'^'^  +  AoMm)x'>'  +  '^  -h  .  •  •    .     (2) 

Equations  may  be  classified  as  binomial^  trinomial,  etc.,  accord- 
ing to  the  number  of  terms  they  contain,  when  written  in  the 
form  (i),  or,  what  is  the  same  thing,  the  number  of  terms  in 
the  result  of  substitution  (2).  Thus,  the  equation  solved  in 
Art.  164  is  a  binomial  equation. 

In  the  general  case,  the  process  of  solving  in  series  is 
similar  to  that  employed  in  Art.  164,  the  form  which  it  is  neces- 
sary to  assume  for  the  series  being 

J^;  ==  t^ArX^'^-^^, 

where  s  is  the  greatest  number,  integral  or  fractional,  which  is 
contained  a  whole  number  of  times  in  each  of  the  quantities  i-„ 
jj,  etc.      As  before,  m  is  taken  to  be  a  root  of  the  equation 


174  SOLUTIONS  IN  SERIES.  [Art.  167. 

/,(;«)  z=  o,  and  Aq  is  arbitrary ;  but,  when  the  coefficient  of  the 
general  term  in  the  complete  result  of  substitution  is  equated 
to  zero,  the  relation  found  between  the  assumed  coefficients 
^01  -^i»  -^a,  etc.,  involves  three  or  more  of  them,  so  that  each 
is  expressed  in  terms  of  two  or  more  of  the  preceding  ones. 
We  can  thus  determine  as  many  successive  coefficients  as  we 
please,  but  cannot  usually  express  the  general  term  of  the 
series. 

We  shall,  in  what  follows,  confine  our  attention  to  binomial 
equations  of  the  second  order. 

Finite  Solutions. 

168.  It  sometimes  happens  that  the  series  obtained  as  the 
solution  of  a  binomial  equation  terminates  by  reason  of  the 
occurrence  of  the  factor  zero  in  the  numerator  of  one  of 
the  coefficients,  so  that  we  have  a  finite  solution  of  the  equa- 
tion.    For  example,  let  the  given  equation  be 

dll^a^l-  21.^0 (I) 

dx'  dx  x"  ^  ' 

This  is  obviously  a  binomial  equation  in  which  s  =.  \  \  hence, 
putting 

7  =  XArX^^r^ 
we  have 

^\\_{fn  ■\-  r){m  -\-  r  ^  i)  —  2\ArX'^  +  r-2 

+  a{m  -f-  r)ArX'^^->'-'^\  =  o. 

Equating  to  zero  the  coefficient  of  ;i:'«  +  ''-2,  we  have 

{ni  j^  r  -\-  \){m  ■\-  r  —  2)Ar  +  a{m  -\-  r  —  i)^r_i  =  o, 

which,  when  r  =  o,  gives 

(«  +  i)(/«  —  2)^0  =  o; (2) 


§  XIV.]  FINITE  SOLUTIONS.  1 75 

and,  when  r>  o, 

The  roots  of  equation   (2)  are  m  =z  —i  and  m  =^  2;  taking 
m  =  —ij  the  relation  (3)  becomes 

Ar  =    -a     ^  ~    ^      Ar.ry (4) 

r{.r  -  3) 
in  which,  putting  r  =  i,  and  r  =  2,  we  have 

A^  =  —a    ~^    Ao, 

l(-2) 

A^  =  —a    ^      A,  =  o. 

2(-l) 

All  the  following  coefficients  may  now  be  taken  equal  to  zero,* 

*  In  general,  when  one  of  the  coefficients  vanishes,  the  subsequent  coefficients 
in  the  assumed  series  I>o  Arx»i  +  rs  must  vanish  ;  in  other  words,  the  value  of  y  can 
contain  no  other  terms  whose  exponents  are  of  the  form  m  +  rs.  But,  in  the 
present  case,  the  assumed  form  \s  y  =.  'Lo  Arxr—  i ;  and  this  includes  the  powers 
x^,  x^ .  .  .  which  we  know  to  be  of  possible  occurrence  since  the  other  value  of  m 
in  this  case  is  2.  Accordingly,  if  we  continue  the  series,  it  recommences  with  the 
term  containing  x'^.     Thus,  putting  r  =  3  in  equation  (4),  we  obtain 

^3  =  -al-A2  =  ?, 
3.0  o 

which  is  indeterminate ;  then,  putting  r  =  4,  5,  etc.,  we  have 

A 4  =  —a  — A3,       As  =  —a  -^  A4  z=  a^^A^,       etc. 
4.1  5.2  4.5 

Thus,  the  assumed  form  j/  =  2©  Arx^—r  really  includes,  in  this  case,  the  complete 
integral 

y  =  aJI  -^)  +  A^'(i  -  ^ax  +  1-a^x^ ±-a^x^  +'\ 

\x       2/  \       4  4.5  45-6  / 


176  SOLUTIONS  IN  SERIES.  [Art.  1 68. 


SO  that  we  have  the  finite  solution  * 

^.;-.  =  ^^-(i  -  2.)  =  ^..(i  -  5). 

169.  For  the  other  solution,  taking  w  =  2,  the  relation  (3) 
becomes 


whence 


Br  =    —a-. ; —Br-i\ 


1.4 
B^  =  -al^B.  =  a^l^B.. 

B,  =  -aA^B,  =  -  a^-^B, 
3.6  4-5-6 


Hence 

Boy^  =  BoX^fi  -  -ax  +  -^a'x' ^a^x^  +  .  .  -V 

\         4  4.5  4.5-6  / 

and  the  complete  integral  is 

2  —  ax  /        2  z  \ 

y  =  Ao h  BoxH  I ax  +  -^a^x^  —  .  .  . ) . 

-^  °      2^       ^     °"  V         4  4-5  / 

170.  Since  we  have,  in  this  case,  a  finite  integral  of  a  linear 
equation  of  the  second  order,  namely, 


*  In  like  manner,  if,  in  a  trinomial  equation,  the  coefficients  between  which  the 
relation  exists  are  consecutive,  a  finite  solution  will  occur  when  two  consecutive 
coefficients  vanish. 


§  XIV.]  EXAMPLES.  177 

equation  (4),  Art.  146,  gives  the  independent  integral 


2X     J     (2  —  ^.::ic)^ 


We  must  therefore  have  y^  =  Ay^  -}-  ^j/j  where  7,  and  y^  are 
the  integrals  found  in  the  preceding  articles,  and  the  constants 
A  and  B  have  particular  values  to  be  determined.  Since  both 
J/'/  and  j/2  vanish  when  ;i;  =  o,  while  y^  does  not,  we  shall  have 
A  =  o;  and,  comparing  the  lowest  terms  of  the  development 
of  the  integral  with  the  series  J2,  we  find  B  =  ^;  hence 

2  —  axt       x^e-^^      7  x^r         2         I      3     2  2  "1 

ax  =  —    I ax  +  -^-a^x^  —  .... 

X      J    (2  -  axy  61        4  4.5  J 


Examples  XIV. 
Integrate  in  series  the  following  differential  equations  :  — 

I.    x—^  +  {x  +  «)-/  4-  («  +  i)y  =  o, 
ax^  ax 

y  =  Afn  —  {n  -\-  i)x  -\-  {n  +  2)—^  —  («  +  3)—  +  .  .  . j 

4-  Bx'-^(l  +  —^—  X  4- r^^ r  X' 

\  n  —  2  («  —  2)(/2  —  3) 


(«  -  2)(«  -  3)(;z  -  4) 


^3  4- 


■> 


2.     ^  4-  ^J'  =  o, 


\     3i       6!        9!  y 

4.  b(x  _  £^^4  4.  ^:x7  -  . .  A 


178  SOLUTIONS  IN  SERIES.  [Art.  I/O. 

^           dx"          dx       ^  ^^ 

y  =  Ax(  I  H H h 


\         2.5        2.4.5.9       2.4.6.5.9.13  / 

\        2.3       2.4.3.7       2.4.6.3.7.1 1  / 


^  x^ 


1.3        1.3.3.7        1.3.5.3.7.11 


4.  ^0+^£  +  ^'^>'  =  2, 


4.  ^/l    _    2-2-^3^3  4.  2^a^x^  -  2^^a9x'^  +  .  .  .Y 
\  5!  8!  11!  J 


^     dx^  -^  * 


y  —  A{\ f-  ■ 1-  ...  I 

\         3-4       34-7-8       3.4.7-8.11. 12  / 

+  Wi  -  ^V  -^!^ ^^^"         +  .  .  .\ 

\         4.5        4.5-8.9       4-5-8-9-12. 13  / 

,    jr*/         ax^    ,       a^x^  \   ,   x^f         ax^   ,        a^x^  \ 

2\         5.6       5.6.9.10  /        6\         6.7      6.7.10.11  / 

6.     X — ^  H-  (^  +  «)—  +  (^  —  i))*  =  ^*-'', 

^  n       i!        ;z  +  i2!       ;2  4-23!  / 

4.  J^lZlf^ L_  £  + I ^  -  .  .  .V 

2  -  n\         S  —  n  2        (3  —  ^)  (4  —  «)  3  / 


§  XIV.]  EXAMPLES.  179 


7-  ^^  +  4^  +  ^  =  °. 


dx^  dx 

y 


V         3i  5I  7!  9  J  / 


8.     ^^+(^  +  2^^)^-4J  =  o, 


\  5  5-6  5-6-7  /         \^       3^       3/ 


9.     (a:  -  ^)-^  +  3^^  +  2;;  =  o, 


y  =  A(6  -  4X  +  x^)  +  ^i-i:A£, 


Show  also  that  x-^(i  —  ic)'»  is  an  integral. 

10.     (4^3  _  14^  _  2;^)^  -  (6J1-2  _  7^  +  i)^ 

^2  ^^j; 


H-  (6^  —  i)y  =  o, 
J  =  Axi(i  H-  2a;)  +  B(i  —  x). 

11.  ^2^  +  ^2^  ^(x-  2)y  =  o, 

dx^  dx 

^  \3       41        52!       63!  / 

12.  Denoting  the  integral  in  Ex.  11  by  Ay^  +  ^j^a,  find,  by  the 

method  of  Art.  146,  an  independent  integral,  and  express  the  relation 

between  the  integrals.  ,  /2    ,        ,      \ 

&  J2  =  e-^(-  ^  2  -\-  x\  =  2y,  -  y^. 

y  =  Ax^fi  +  ^  -f  -^  +  -^  +  .  .  A  +  ^/l  +  I  +  £\ 
\         4       4.5        4.5-6  /  \^  2/ 

Show  also  that  x-^e^  is  an  integral. 


l80  SOLUTIONS  IN  SERIES.  [Art.  I/O. 

14.     ^Ui  -  4A)yi  +  [(I  -  n)x  -  (6  -  4«)^]^ 

+  «(l  —  «)^J  =  o, 

A  J      X           i    ^(^  +  3)    3       «(«  +  4)(«  +  5)-  ,    ,         \ 
y  =  ^:r«(  I  +  «a:  H j^ ^^  H — j ^3  +  .  .  .  j 

^  j,f                ,    ^(^  -  3)    3       ;2(;2  -  4)(«  -  5)    ,    ,         \ 
+  JB\^i  -  nx  +  -^ X' : —^ ^3  +  .  .  .j. 

y  =  Ax-^li  -  -^  +  — W  ^^^/'i  -  -^x  +  -4:5_^2  _  .  .  .\ 
\         5  20/  V         1-7  1-2.7.8  I 

16.  (a^  +  :v^)— ^  +  ::c-=^  —  n^y  =  o, 

^       \      2\a^  4!      «4^     y 

17.  Denoting  the  integral  given  in  Ex.  16  by  Ay,  +  By2,  show 

and  find  the  corresponding  result  when  «  =  o. 

log  [x  +  v/(^^  +  ^^)]  =  log^  +  ^-i-^  +  ^  —  -... 

a       2  3^3        2.4  5^5 

18.  Expand  sin(^sin-' jc)  and  cos  («cos-':r)  by  means  of  the 
differential  equation  ,3  , 

('  -  "^>5^  -  "i  +  '^'^  =  °' 

of  which  they  are  independent  integrals, 
sin  («  sin- '  ^)  =  axil 1—  x^  +  ^^ —^ ^  x^  ^  ,  ,  .h 

cos(tf  sin-  ^jc)  =  I  -  ^x^  +  flifi-ZUtl^  -  .  . . 
2  1  4  ! 


§  XV.]  CASE   OF  EQUAL    VALUES  OF  m.  l8l 

XV. 

Case  of  Equal  Values  of  m, 

171.  If  the  two  roots  of  the  equation  determining  m  are 
equal,  we  can  determine  one  integral  of  the  form  y  =  ^ArX'"  +  ^^ 
by  the  process  given  in  the  foregoing  articles  ;  but  there  is  no 
other  integral  of  this  form.  We  therefore  require  an  independ- 
ent integral  of  some  other  form. 

For  example,  let  the  given  equation  be 

^^'  ""  ""'^S  "^  ^'  ~  ^^^^fx  -  ^->'  =  °>  •     •     •     (') 

a  binomial  equation,  in  which  we  may  take  s  =  2,  or  s  =  —2. 

Assuming  ^«,  ^ 

we  have,  by  substitution, 

2~[(;/2  +  2ryArX»'  +  ^^--'  —  {m  +  2r  -\-  iyArX^'  +  ^^+^2  =  o. 

Equating  to  zero  the  coefficient  of  z"'  +  ''''-^,  we  have 

(m  -f  2ryAr  —  (m  -\-  2r  —  lYAr-x  =  0..     .     .     (2) 

Putting  r  =.  Oy  m^Ao  =  o  ;  whence 

m  =  o, 

the  two  values  of  m  being  identical.    Putting  7^  =  o  in  equation 
(2),  the  relation  between  consecutive  coefficients  is 

_  {2r-  lY  ,       . 

^''"     {2ry    ^"-^^ 

whence  we  find  the  integral 

(-.2  1-2    ,2  -2    ,2    ^2  \ 

2^  2^4^  2^.4^.6^  / 


1 82  SOLUTIONS  IN  SERIES.  [Art.  1 72. 

172.  To  obtain  a  new  integral,  we  shall  first  suppose  the 
given  equation  to  be  so  modified  that  one  of  the  equal  factors 
in  the  first  term  of  equation  (2)  is  changed  to  w  +  2r  —  //,  so 
that  one  of  the  values  of  m  becomes  equal  to  //,  while  the  other 
value  remains  equal  to  zero.  We  shall  then  obtain  the  complete 
integral  of  the  modified  equation,  in  which,  after  some  trans- 
formation, we  shall  put  h  =  o,  and  thus  obtain  the  complete 
integral  of  equation  (i). 

The  altered  relation  between  consecutive  coefificients  may 
be  written 

{m  -\-  2r){m  -\-  2r  —  h) 

in  which,  for  a  reason  which  will  presently  be  explained,  //  is 
put  in  the  place  of  h.     Hence,  when  tn  =  o,  we  have 

and  the  first  integral  now  is 

>'•='+  -T^-T^^'  +  —f T^TT FT^'  +  •  •  •  •   (5) 

2{2  —  h)  2.4(2  — /?  )(4  —  ^  ) 

Putting  m  =^  km  equation  (4),  we  have 

B   =  (2r  -  I  +  hy        jg 

{2r  +  h){2r  -  h' +  h)     ''"'' 

and  the  second  integral  is 

^'  \  (2    +   h){2    -   h'  +   h) 

+ (I  +  hyii  +  hY ^  +  .,.\    (6) 


§  XV.]  CASE   OF  EQUAL    VALUES  OF  m.  1 83 

if, 

The  object  of  introducing  //  in  equation  (4),  in  place  of  the 
equal  quantity  //,  is  that,  when  equation  (6)  is  written  in  the 
form 

\\iiji)  shall  be  such  a  function  of  h  that,  by  equation  (5), 
.    ,  y,^  V'(o)- 

Developing  y^  in  powers  of  //,  we  have,  since  x^  =  ^^^s* 
^^3  =  (i  +  h\ogx  +  . .  .)[;'i  +  #'(0)  +  ...]; 
hence  the  complete  integral  is 

y  =  Aoy,  +  Boy.  +  BJi\_y,\ogx  +  r\,\6)  +  ...]; 
or,  replacing  the  constants  A^  +  B^  and  BJi  by  ^  and  B^ 

y  =  4);,  +  By, \ogx  +  ^i/.'(o)  -}-...,...     (7) 

in  which  we  have  retained  all  the  terms  which  do  not  vanish 
with  h,  and,  when  h  =  o,  y,  resumes  the  value  given  in 
equation  (3). 

173.  It  remains  to  express  y\ii^  in  terms  of  x.  In  doing 
this,  we  may,  since  //is  finally  to  be  put  equal  to  zero,  make 
this  substitution  in  the  value  of  \\i{Jt)  at  once,  and  write 

^^  ^  (2  +  hy     ^  (2  H-  /0H4  -f-  hy     ^  ^  ^ 

Denote  the  coefficient  of  x'^^'m.  this  series  hy  Hr,  so  that  H^^^  it 
and  when  r  >  o, 

H   =  (I  +/^)-(3  +  >^)-...(2r-  I  +hy  .  ,  . 

(2  4-^)^(4  +  /^)-...(2r  +  y4)^      .    •     •     vyy 


1 84  SOLUTIONS  IN  SERIES.  [Art.  1 73. 


then 
and 

^{h)^X'^-^X^r^XH/-:^^X^r^.      .      .       (10) 

an  an 


JTT 

in  which  unity  is  taken  as  the  lower  limit  because  — — °  =  o. 

ah 


From  equation  (9), 

d\ogHr  _      2  2        I   _ , 


+ 


dh  \  •\-  h       3  +  ^  2r—  1+^ 


2  -\-  h       ^  +  h  2r  -{-  h* 

which,  when  ^  =  o,  becomes 

dXogHjT]    _  2       2  2  22  2 

^'^      Jo        ^        3  2/*  —  I        2       4  2r 

whence,  putting  //  =  o  in  equation  (10),  and  denoting  i//'(o), 
when  thus  expressed  as  a  series  in  x,  by  y, 

Hence,  when  //  =  o,  equation  (7)  gives  for  the  complete  integral 
of  equation  (i)* 

y  =  Ay,  +  BiyAogx  +  y'), 

where  y,  and  _^'are  defined  by  equations  (3)  and  (11). 

*  For  the  complete  integral  when  we  take  s  —  —2,  see  Ex.  XV.  7. 


§XV.]       INTEGRALS  OF  THE  LOGARITHMIC  FORM,  1 85 


Case  in  which  the  Values  of  m  differ  by  a  Multiple  of  s. 

174.  When  the  two  values  of  m  differ  by  a  multiple  of  j,  the 
initial  term  of  one  of  the  series  will  appear  as  a  term  of 
the  other  series  ;  and  the  coefficient  of  this  term  will  contain 
a  zero  factor  in  its  denominator.  Hence,  unless  a  zero  factor 
occurs  in  the  numerator,*  the  coefficient  will  be  infinite ;  and, 
as  in  the  preceding  case,  it  is  impossible  to  obtain  two  inde- 
pendent integrals  of  the  form  ^ArX'^^''\  For  example,  let  the 
given  equation  be 

^2(x  4.  oc)p-  +  ^^  +  (I  -  2x)y  =  o.   .    .    .     (i) 
dx^  dx 

Putting  y  =  A^x"^  in  the  first  member,  the  result  is 

Aoiur"  +  i)x'''  4-  Ao{m^  —  m  —  2)x'''  +  ^. 

Choosing  the  second  term  as  that  which  is  to  vanish  by  the 
determination  of  m,  because  the  first  would  give  imaginary 
roots,  we  have 

rn  —  —\     or     m  —  2,     and    ^  =  —  i ; 

hence,  putting  y  —  '^^A^x"'-'', 

lo\{m  —  r  -\-  i){m  —  r  —  2)ArX»'-^  +  '' 

+  [(;/z  —  ry  H-  i^ArX^'-^l  =  o; 

and,  equating  to  zero  the  coefficient  of  x'"-'^  +  '', 

{m  —  r-\-  i){m  —  r  —  2) A  +  [(^^  —  r  +  i)^  +  i]^r-i  =  o.   (2) 


*  It  is  immaterial  whether  the  zero  factor  in  the  numerator  first  occurs  in  the 
term  in  question,  or  in  a  preceding  term  ;  the  result  is  a  finite  solution.  An  example 
of  this  exceptional  case  has  already  occurred  in  Art.  168,  where  s  =  i,  and  the 
values  of  m  differ  by  an  integer. 


1 86  SOLUTIONS  IN  SERIES.  [Art.  1 74. 

When  ;;/  =  —!,  the  relation  between  consecutive  coefficients  is 

r{r  +  3) 
and  the  first  integral  is 

Aoy^  =  AoX-^i —x-^  H ^^^x-' 

\         1.4  1.2.4.5 


1.2.3.4.5.6  / 


Putting  m  =  2,  the  relation  is 


and  the  second  integral  takes  the  form 


Boy2  =  Box^fi 5_  ^-i  + _5^ 

\  —2.1  — 2(  — 1).1.2 

5>2.i 


2(— l).O.I.2.3 


.),    (4) 


in  which  the  coefficient  of  x-''  is  infinite.  Thus,  the  second 
integral  of  the  form  ^A^x*^-^''^  fails,  and  we  require  an  inde- 
pendent integral  of  some  other  form. 

175.  To  obtain  the  new  integral,  we  proceed  as  in  Art.  172. 
Thus,  supposing  the  second  factor  in  the  first  term  of  equation 
(2)  to  be  changed  to  m  —  r  —  2  —  h,  so  that  the  second  value 
of  m  is  now  2  +  h  instead  of  2,  and  using  //  as  in  Art.  172,  the 
relation  between  consecutive  coefficients  now  is 


A    - {m  -  r  +  lY  +  1  .  .  . 


§  XV.]        INTEGRALS  OF   THE  LOGARITHMIC  FORM.  1 8/ 

When  w  =  —  I,  this  becomes 

^(^  +  3  +  ^  ) 
and  we  have 

Aoy^  =  Aox-Ui -—^——x-' 

+  £i5 _^-2  _  .  .  \      (6) 

i.2(4  +  /^0(5  +^')  /       ^ 

Putting  m  =z  2  -^  k,  the  relation  between  the  coefficients  in  jj 
Br= (r-^-ky  +  i        B 

and  the  new  value  of  B^y^  is 

\      (-2  -  /^)(i  4-  /^'-  /^)  y 

in  which  the  first  term  which  becomes  infinite  when  ^  =  o  is 

^  [(-2  -  hf  +  I]  [(-I  -  hf  +  i][(-/^r+  I]  ^-1  +  ;^      /_x 

Denoting  the  coefficient  of  this  term  by  —,  and  the  sum  of  the 
preceding  terms  in  y^  by  7",  we  may  write 

B^y,  =  B,T 

,   B  J  (i  -hy  ^T  \ 

If  now  we  write  this  equation  in  the  form 

Boy^  =  BoT  +  ^xhxp{h), 
n 


1 88  SOLUTIONS  IN  SERIES.  [Art.  1 75. 

equation  (6)  shows  that  /,  =  1/^(0) ;  hence  the  complete  integral 
may  be  written 

y  =  ^ojF.  +  B,T  +  |(i  -h  ^logx  4- . .  .)[jx  +  ^'A'(o)  +  ...]» 
or,  putting  A  for  the  constant  ^o  +  y » 

y=.  Ay,  -h  BoT  +  ^^'.log^  +  ^f  (o)  + (9) 

In  this  equation  we  have  retained  all  the  terms  which  do  not 
vanish  with  h_\  from  the  value  of  By  as  defined  by  the  expres- 
sion (7),  we  see  that,  when  h  =.  Oy 

B^B  -^ =  ^^0;.    .    .    .     (10) 

(_2)(-i). 1.2.3       6 

and,  when  ^  =  o,  we  have,  from  equation  (4), 

T  =  :x^  -\-  ^x  +  i (11) 

176.  The  expression  for  i/^'(o)  as  a  series  in  x,  which  we 
shall  denote  by  j',  is  found  exactly  as  in  Art.  173.  Putting 
^ '  =  o  at  once,  in  the  value  of  il/{/i)  as  defined  by  equation  (8), 
we  have 


^^   '  V         (I  -A)U^k) 


(I  -/i){2  _/0(4-/0(5  -^)  / 

and,  writing  this  in  the  form 

we  have  Ho  =  i,  and,  when  r>  i, 

H  =  r(i  -  hY  +  iir(2  -hy  +  x^... \(r - hy  +  11 

(I  -  A) (2  -  /i)  .  . .  (r  -  /5)(4  -  /5)  .  .  .  (r  +  3  -  /i) 


§  XV.]       INTEGRALS  OF  THE  LOGARITHMIC  FORM.  ll 


Hence 


in  which 


^\h)  =  X-^X{-^yHj^-^^^^X-r,      .       .       .       (12) 


dXogHr  ^  2(1  -  h) 2(2  -  h)       __ 

dh  (1-/^)^  +  1        {2  -  hy  +  1 

(^r  ^  hy  +  1        1  -  h       2  -  h 

r  —  h       4  —  k  ^  +  3 

When  /i  =  o,  this  becomes 
Y/lOg^r"]   _.  _  __2 4__  _  _  ^  _ 


^A     J  12  +  1       2^  +  1  r'  +  I 


i  +  I  +  ...  +  I+i  +  £  +  ...+      ^ 


12  r       4       5  ^  +  3 

hence,  putting  /^  =  o  in  equation  (12),  we  have 

L1.4V2      I      4/ 

»  _JJ_/?  +  4  _  I  ^  I  _  i  ^  i\^_,  +  .  .   ~ 
1.2.4.5V2        51245/ 


(13) 


Now,  putting  ^  =  o  in  equation  (9),  substituting  Bo  =  ^B 
from  equation  (10)  and  the  value  of  T  from  equation  (11),  we 
have,  for  the  complete  integral  of  equation  (i), 

y  =  Ay,  +  Bilx"  +  3^  +  3  +  ;;,log^  +  y'), 

where  y^  and  y'  are  defined  by  equations  (3)  and  (13). 


1 90  SOL UTIONS  IN  SERIES.  [ Art .   l^f^ 


Special  Forms  of  the  Particular  Integral. 

177.  We  have  seen,  in  Art.  166,  that  the  particular  integral? 
when  the  second  member  of  the  given  equation  is  a  powel 
of  X,  may  be  expressed  in  the  form  of  a  series  similar  to  those 
which  constitute  the  complementary  function.  Special  cases 
arise  in  which  the  particular  integral  either  admits  of  expression 
as  a  finite  series,  or  can  only  be  expressed  in  the  logarithmic 
form  considered  in  the  preceding  articles.  In  illustration,  let 
us  take  the  equation 

(^--)g-|  =  ^-' (') 

of  which  the  complementary  function  is  A  sin-^;ir  +  B.    Putting 
y  =  2"^r^'«  +  =^  we  have 

^Ar\^{m  -f-  2r){m  +  2r  —  i)x^^^^-'^ 

—  (»/  +  2ryx^^^^'\  =  px'^'i     (2) 
whence,  when  ^  >  o, 

{m  +  2r){7n  -}-  2r  —  i)Ar  —  {m  -{-  2r  —  2)^Ar-i  =  o, 

and  the  relation  between  consecutive  coefficients  is 

{m  +  2r){m  -j-  2r  —  i) 

For  the  complementary  function,  we  have  m  =  i,  or  m  =  o. 
Putting  m  =:  I  in  equation  (3), 

.    _    (2r  -  i)^  . 

2r{2r  4-  i) 
whence 

'  +  i:3^  +  5:ili-^  +  ---)    •  •  •   (4) 


§  XV.]   SPECIAL  FORMS  OF  THE  PARTICULAR  INTEGRAL,     191 

This  is  the  value  of  sin-';tr.    The  series  corresponding  to  w  =  o 
reduces  to  a  single  term,  so  that 

yz  =  I. 
For  the  particular  integral  F,  we  have,  from  equation  (2), 

whence 

m  =  a  +  2,    and     Aq  = 


(a  +  i)(«  +  2) 
Putting  m  =  a  +  2mthe  relation  (3), 

yt    __  (^  +  2ry .       , 

{a  +  2r  +  i){a  -^  2r  +  2) 
hence 

(«+    !)(«+   2)V  (d!   +   3)(«  +  4) 

+ (^  +  2)-(^  +  4)- ^  +  ...Y      (5) 

(«  +  3)(«  +  4)(«  +  5)(«  +  6)  / 

This  equation  gives  the  particular  integral  except  when  a  is 

a  negative  integer ;  for  instance,  when  ^  =  o,  and  /  =  2,  it 

gives 

Y  =  x^li  +^x^  +  -^^x^  +  ...Y 
\        34  34-5-6  / 

which,  as  will  be  found  by  comparing  the  finite  solution  of 
equation  (i)  in  the  case  considered,  is  the  value  of  (sin-';l^)^ 

178.  Now,  in  the  first  place,  if  «  is  a  positive  odd  integer, 
all  the  powers  of  x  which  occur  in  F  occur  also  in  ji ;  and,  when 
this  is  the  case,  we  can  obtain  a  particular  integral  in  the  form 
of  a  finite  series.     For  example,  if  <3:  =  3,  we  have 

K  =  ^(i+f^  +  /^^  +  ...Y 

4.5  V      6.7       6.7.8.9  / 


192  SOLUTIONS  IN  SERIES.  [Art.   1/8. 

If  we  write  this  equation  in  the  form 

2.3/       2.3.4.5  \      6.7  / 

the  second  member  is  equivalent  to  the  series  j„  equation  (4), 
with  the  exception  of  its  first  two  terms.     Thus 

^  =  ..-(.  +  i4     or     y  =  ^,.-^(.  +  I.3), 

and,  since  the  first  term  of  this  expression  is  included  in  the 
complementary  function,  we  have  the  particular  integral 

Y  = X x^, 

3  9 

This  finite  particular  integral  would  have  been  found  directly 
had  we  employed  a  series  in  descending  powers  of  x. 

179.  In  the  next  place,  when  «  is  a  negative  odd  integer, 
the  initial  term  of  ^,  will  occur  in  Y  with  an  infinite  coefficient. 
Thus,  if  ^  =  —3  in  equation  (5),  Art.  177,  the  second  term 
contains  the  first  power  of  x  and  has  an  infinite  coefficient. 
To  obtain  the  particular  integral  in  this  case,  suppose  first  that 
^  =  —3  ■\-  h\  then  equation  (5)  gives 

y= /^-^^^ 

(-2 +/^)(-i  +/^) 

/(-I  ^-hyx^^f^        I  (I  ^hy     ^         \ 

Putting 


§  XV.]    SPECIAL  FORMS  OF  THE  PARTICULAR  INTEGRAL.     1 93 

equation  (4),  Art.  177,  shows  that  j,  =  »/^(o) ;  and  we  may  write 
y  =  r  -f-  ^(i  +  h\ogx  +  .  .  .)[jx  +  h^\,\6)  +  .  .  .] 

.here  TV  is   a  quantity  which   remains  finite  when  h  =1  o. 
Expanding,  and  rejecting  the  term  -y-Ji,  which  is  included  in 

the  complementary  function,  we  may  now  take,  for  the  particular 
integral, 

r  =  T  +  NyAogx  +  iVV'(o)  4-  .  .  .  , 

in  which  we  have  retained  all  the  terms  which  do  not  vanish 

P  P 

with  h.     When  h  =  o,  the  values  of  T  and"  JV  are  —  and  - 

2X-  2 

respectively ;  and,  finding  the  value  of  «/''(o),  as  in  Arts.  173 
and  176,  we  hav^,  for  the  particular  integral, 


1^.3^   /2  2  I  I  I  l\  "1 

+  ^7  +  5-i-3-4-5>+---J- 

180.  In  like  manner,  when  a  is  a,  negative  even  integer, 
the  term  containing  x°,  corresponding  to  j/j,  occurs  in  V  with 
an  infinite  coefficient.  Thus,  if  ^  =  — 4,  the  second  term  of 
the  series  in  equation  (5),  Art.  177,  is  infinite.  But,  putting 
a  =  — 4  +  //,  we  have 

V  =  /-^-'"^  .  /(-2  +  /lY 

(-3  +  ^){-2  +  h)       (-3  +  A)(-2  -h  /i){-i  +  h)k 

\  (i  -j-  h){2  -h^)  / 


or 


K=  T+^{i  -h/i\ogx  +  ...)xl^{h). 


194  SOLUTIONS  IN  SERIES.  [Art.  l8o. 

In  this  case,  when  \\i(Ji)  is  expanded  in  powers  of  //,  the  first 
term  is  unity,  and  there  is  no  term  containing  the  first  power 

of  // ;   hence,  rejecting  the  term  —  which  is  included   in  the 

complementary  function,  and   then   putting  h  =■  o,  we  have 
the  particular  integral 

6x'        3  ^°^'^* 


Examples  XV. 
Integrate  in  series  the  following  differential  equations  :  — 

ax^       ax 
y  =  (A  +  B\ogx)(j  -  ^  +  ^ ^i_  +  .  .  \ 

rt^v    ,    dy   \    ^ 

L   2^  2^4^  \  2/  2^4\6^V  23/  J 

V       1.2      I.2^3      I.2^3^4  / 

^B  +  ^^cf^  A  _   J^U  +  ^U  .  .  .1 

|_1.2    1.2  I.2^3\I.2  2.3/  J 


§xv. 


EXAMPLES. 


195 


4.     ^3^  -  (2^  -  i)y  =  o, 


\  1.4        1.2.4.5        1.2.3.4.5-6  / 

+  3^(4^^  +  2a:  +  I) 

L1.4V      4/     1.2.4.5  \      245/         J 


^*    "^'^  "^  "^^^  "^  ^^^  "^  ^'^'^  ~  ^^-^  ^  ^' 


\         3  I       32  1      335  /       3 

+  ^X4£/I  _  I  +  A  _  5^/i  _  i  +  ,  +  iW  .  .  1 
L3  A3       4         /        3  2  \3       5  2/  J 


6.     (x  —  x') — ^  —  J  =  o, 

dx^ 


y  =  (A  +  B\ogx)x(i  +  —X  +  -^^'  -I ^^^  ^3  +  .  .  . ) 

•^       ^  °    ''    \     '    1.2  i.2''.3  1.22.32.4        '         / 

Ll.2\I  I  2/        ^    I.22.3\l    ^3  I  2  3/ 

7.     Find  the  integral  of 

^^(i  _  ^2)  J  +  (i  -  3^')-r  -  ^>'  =  o, 
[equation  (i),  Art.  171,]  when  x  >  1. 
y={A  +  B\ogx)x-{i  +  ^x--  +  ^x-^  +  .  .  .^ 

\  2^  2^.42  / 


—  2Bx- 


_2'   1.3  2^4=\l.2  3.4/  J 


196  SOLUTIONS  IN  SERIES.  [Art.  180. 

8.  ^  +  2f  =  o, 

(Aax^        A*a*x  A^a^x^  \ 

[AaxUi        i\        A^a^x  Iy        i        i        i\  l 

9.  ;e^  -  (a:'  +  4^)  ^  +  4j>,  =  o, 

CA"^  ax 

y  =  ^;c*tf-»^  4-  -^(2.r  —  ;c2  +  .^3  4.  ^cV^logx) 
10.     xii  -  A-)g  +  (I  -  A-)£  -irxy  =  o, 

^2^4^6'\4       6       3       5/  J 

>  =  (^  +  ^log.)^(x  +  £.  +  ^^  +  ^^,-^  +■■■) 

-^(3.-8.)+.5.[5(i-l.-i). 

+ -i!:5l/2  _  1  _  I  +  2  _  1  _  lU  +  . .  I 
^  2.4.6.8V3      26^548/  J 


§  XV.]  EXAMPLES.  197 

12.      x^—^  -j-   y  =  jvt, 

y  =  (A  +  B\ogx)(i  -"^  +  -^  -  -4^  +  .  .  .) 
\         1.2        I.2^3        I.2^3^4  / 

L1.2V1       2/       i.2^3\i       23/  J 

V  1-3         1-3^5        1.3^5^7  / 


8     _i 
— X    2 

25 


105       \ 


2       ,  ,      2.4 

I  —  — x-^  -\ -i — X- 


3.9  34-9-II  / 

14.     Express  the  particular  integral  of  the  equation 

(a)  in  the  form  of  an  ascending  series ;  (/?)  in  the  form  of  a  descending 
series ;  (y)  as  a  finite  expression.  [See  Example  XIV.  9,  for  the 
complementary  function.] 

W  y  =  -(-^^log.  +  5-  -  ^^  +  f .-  - '-!.-' 

5  V     6        6.7  y 


198  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 8 1. 


CHAPTER    VIII. 

THfe  HYPERGEOMETRIC   SERIES. 

XVI. 

\ 

General  Solution  of  the  Binomial  Equation  of  the  Second  Order, 

181.  The  symbol  i^(a,  /?,  y,  z)  is  used  to  denote  the  series 

^    ^   gjg^   ^   a(a4.l)^(^+l)^^    ^    a(a-|- 1)  (a  +  2)^(^+ 1)  (/?4-2)^,  ^    ^^^ 
i.y  l.2.y(y+i)  i.2.3.y(y+i)  (y+2) 

which  is  known  as  the  hypergeometric  series.  Regarding  the 
first  three  elements,  a,  yS,  and  y,  as  constants,  and  the  fourth  as 
a  variable  containing  x,  the  series  includes  a  great  variety  of 
functions  of  x.  In  fact  we  shall  now  show  that  one,  and  generally 
both,  of  the  independent  integrals  of  a  binomial  differential 
equation  of  the  second  order  whose  second  member  is  zero  can 
be  expressed  by  means  of  hypergeometric  series  in  which  the 
variable  element  is  a  power  of  x. 

182.  Using  the  notation  of  Art.  123, 

X—  =  ^,  whence  x"—  =  ^(^  -  i), 

dx  dx"  ^  ' 

we  may,  as  in  Art.  167  (first  multiplying  by  a  suitable  power 
of  x)y  reduce  the  binomial  equation  to  the  form 

/(^)7  4-  x^^{{^)y  =  0, (i) 


§XVI.]    BINOMIAL   EQUATION  OF  THE  SECOND  ORDER.    1 99 


in  which  /  and  <^  are  algebraic  functions,  one  of  which  will  be 
of  a  degree  corresponding  to  the  order  of  the  equation,  and  the 
other  of  the  same  or  an  inferior  degree.  If  the  equation  is  of 
the  second  order,  it  may  be  written 

(^  _  a)  (O  -  d)y  -  ^x'{h-  —  c)  {&  -  d)y  ^  o,    .     .   (2) 

in  which  q  and  s  are  positive  or  negative  constants.  Further- 
more, the  equation  is  readily  reduced  to  a  form  in  which  q  and  s 
are  each  equal  to  unity ;  for,  putting 

d 


we  have 


z  =  qx^,         and  ^'  =  _ 

dz 


^'  =  ^x' ^ =  i^,         or         &  =  sd-' 

qsx^  -  ^dx       s 


and,  substituting,  equation  (2)  becomes 

{*'-7){^'-7)--<^--:)(^-7>=°--  -(3) 

183.    We  may,  therefore,  suppose  the  binomial  equation  of 
the  second  order  reduced  to  the  standard  form 

(t9-  -  a){d-  -  b)y  -  x{d-  -  c){&  -  d)y  =  o.      .     .   (i) 

Substituting  in  this  equation 

y  =  %'^ArX^  +  ^, 
we  have 

^"^Arl^m  +  r—a)  {m-\-r—d)x^+'>'—{m-\-r—c)  (rn-\-r—d)x»f+*'+^']  =0, 

and,  equating  to  zero  the  coefficient  of  x^  +  % 

{m-\-r—a)(m-\-r—b)Ar—{m-{-r—i—c){fn-\-r—i—d)Ar-i  =  o. 


200  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 83. 

This  gives  the  relation  between  consecutive  coefficients, 

and,  when  r  =z  o, 

(m  —  a){m  —  b)Ao  =  o ; 

whence  m  =  a  or  m  =z  b.     Putting  m  =.a,vft  have  for  the  first 
integral 

>'  =  -"('  +  V^iVf^ 

\  i{a  -  b  -{-  1) 

^  i.2{a-b->ti){a-b+2)  ^  +  ---j»    W 

and,  interchanging  a  and  b,  the  second  integral  is 

\  i{d  -  a  +  1) 

^^  -  c){b  -  c  +  i){b  -  d){b  -  d  +  1)  \      .. 

■^  i,2{b-a+i){b-a  +  2)  T.-.y.    u; 

Thus,  putting 

a  ^  c  =  a  1 

^  -  ^  =  ^     , (4) 

«  —  <5  +  I  =  7  J 
the  first  integral  is 

\          i.y                1.2.7(7  +1)                         / 
=  :^i^(a,A7,^), (5) 

and  the  second  may  be  written 

y,  =  a^Fip!,  ft',  Y,x), (6) 


§  XVI.]    DIFFERENTIAL  EQUATION  OF  THE  SERIES.  201 

where 

a   =  b  —  c  =a-|-i—  y 

P'  =  b-d  =  /3  -f-  I  _  y  [  ,  .    .    .    .    .  (7) 

y'=ib  —  a-\-i=  2  —  7 

and 

b  =  a  -{•  1  —  y. 


Differential  Equation  of  the  Hypergeometric  Series. 

184.  If  in  equation  (i)  of  the  preceding  article  we  put  ^  =  o, 
and  introduce  a,  y3,  and  y  in  place  of  b^  c^  and  d  by  means  of 
equations  (4),  we  obtain 

^(^  -  I  +7)j-:v(^  +  a)(^  +  y8);;  =  o,  .     .     .   (i) 

or,  since  %  =  x—-  and  &^  =  x^— [-  ;r-— ,  in  the  ordinary  no- 

dx  dx^         dx 

tation 

A:(i-^)gH-[y-^(i+a  +  «]£-ay8);=o/.      .    (2) 

This  is,  therefore,  the  differential  equation  of  the  hypergeometric 
series,  Fia,  y8,  y,  x).  Putting,  also,  ^  =  o  in  the  value  of  y^^  we 
have 

;;  =  AF{a,l^,y,x)  +  Bx'-yF{a  -f  i  -  y,  ^  +  i  _  y,  2  -  y,  ^) 

for  the  complete  integral  of  equation  (2). 

Since  the  complete  integral  of  the  standard  form  of  the  bi- 
nomial equation  of  the  second  order,  (i)  Art.  183,  is  the  product 
of  this  complete  integral  by  x"^,  it  follows  that  the  general 
binomial  equation  of  the  second  order,  equation  (2),  Art.  182, 
is  reducible  to  the  equation  of  the  hypergeometric  series  in  v 
and  2  by  the  transformations  ^  =  ^x^  and  j/  =  z^v. 


202  THE  HYPERGEOMETRTC  SERIES.  [Art.  1 85. 


Integral  Values  of  y  and  y\ 

185.  When  a  ■=■  b  \n  equation  (i),  Art.  183,  y  =  y'=  i,  and 
the  integrals  y^  and  y^  become  identical,  so  that  there  is  but  one 
integral  in  the  form  of  a  hypergeometric  series.  Again,  if  a 
and  b  differ  by  an  integer,  one  of  the  series  fails  by  reason  of 
the  occurrence  of  infinite  coefficients.  In  this  case,  let  a  denote 
the  greater  of  the  two  quantities,  then  y  is  an  integer  greater 
than  unity,  and  y'  is  zero  or  a  negative  integer. 

The  coefificient  of  x*'-^^  in  F{a!y  ^,  y\  x),  is 

(g  +  i-y)  .  ♦ .  (g  +  ^-i-y)  (/3+1-7)  •  -  (jg  +  ^-i-y)^ 

{n-i)\{2-y){s-y)  ...{n-y) 

This  is  the  coefficient  oi  x^  +  ''-^,  that  is,  oi  x^  +  '^-y  iny^,  and  is 
the  first  which  becomes  infinite  when  y  =  n.     Now,  putting 


and  denoting  the  sum  of  the  preceding  terms  of  jj  (which  do 
not  become  infinite  when  k  =  o)  by  T,  the  complete  integral 
may  be  written 

in  which  —  is  the  product  of  B^  and  the  coefficient  written  above, 
/I 

so  that,  when  /i  =  o,  B  has  the  finite  value 

B  =  B  (a-hi-n)...(a-i){ft+i-n)...(^-i)       .  . 
{n-i)\{2-n)is-n)..,(-i)  "^  ^ 

Putting 


§  XVI.]  INTEGRAL    VALUES  OF  y  AND  y'.  203 

we  have,  as  in  Arts.  172  and  175,  j^j  =  f(o)j  and,  expanding  in 
powers  of  h,  equation  (i)  becomes 

y  =  Aoy,  +  BoT+  |(i  +h\ogx  +  ...)[;;,  +  hxl;' {p)  +...]; 

or,  putting  A  for  A^  +  y  and  j'  for  ^\o), 

y=^Ay,+BoT+ByAogx  +  By'-^...,.     .     .(4) 

in  which  we  have  retained  all  the  terms  which  do  not  vanish 
with  /i. 

To  find  ^'  or  xf/\o),  we  have,  from  equation  (3), 

ah 
whence,  putting  /^  =  o, 

'•  =  -[r?(M-T- ;)-■■■]■ 


(5) 


Finally,  writing  the  complete  integral  (4)  in  the  form 
y  =  Ay^  +  B-T],  and  taking  the  value  of  B^  from  equation  (3), 
we  have,  for  the  second  integral, 

(a+l-7)...(a-i)(^+l-7)...(^'-l) 

where  y^  is  the  first  integral  x'^F{a,  /?,  y,  x),  T  the  terms  which 
do  not  become  infinite  in  the  usual  expression  for  the  second 
integral,  and  j^' the  supplementary  series  given  in  equation  (5). 

It  is  to  be  noticed  that  when  y  =  i,  T  =  o. 

186.  In  this  general  solution  of  the  case  in  which  y  is  an 
integer,  the  supplementary  series  y'  is  the  same  as  the  first 


204  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 86. 

integral  j„  except  that  each  coefficient  is  multiplied  by  a  quantity 
which  may  be  called  its  adjunct.  The  adjunct  consists  of  the 
sum  of  the  reciprocals  of  the  factors  in  the  numerator  diminished 
by  the  like  sum  for  the  factors  in  the  denominator.  The  first 
term  in^^  must  be  regarded  as  having  the  adjunct  zero. 

If  y^  is  a  finite  series,  it  is  to  be  noticed  that  the  adjunct  of 
each  of  the  vanishing  terms  is  infinite  and  equal  to  the  reciprocal 
of  the  vanishing  factor.  Thus  the  corresponding  terms  of  the 
supplementary  series  do  not  vanish,  but  are  precisely  as  written 
in  the  expression  for  ^„  except  that  the  zero  factors  in  the 
numerators  are  omitted. 

187.   As  an  illustration,  let  us  take  the  equation 

^""^ "  ^'^S+ ^^"^^  2 "  ^'  ~  ^^^-^ "  °' 

which,  when  written  in  the  form  (i).  Art.  183,  is 
(^  -  i);'  -  oc{^'  -  9)y  =  o, 

so  that  ^  =  I,  ^  =  —  I,  ^  =  3,  <^  =  — 3  ;  whence  a  =  —2,  )8  =  4, 
y  =  3.     We  have,  therefore, 

\  1-3  1.2.3.4         / 

2.  —  1.0.A.C.6    ./      ,    1.7      ,         \  /  \ 

1.2.3^4.5       V      4.6         / 


+  -^- 


in  which  the  terms  following  the  first  three  vanish.  For  the 
other  integral,  employing  equation  (6),  Art.  185,  because  y  is 
an  integer,  we  have 

V  =  yi^ogx x-'(i  +  —^x]  +  y', 

-4.-3.2.3      \        -I.I  / 


§  XVI.]  IMAGINARY  VALUES  OF  a  AND  /?.  205 

where  the  next  term  in  the  expression  for  T  would  be  infinite. 
The  part  of  y'  corresponding  to  the  actual  terms  of  j,  is 

L    1.3    \       2       4  3/  1.2.3       \        2  5  2        3/       J 

and  the  part  corresponding  to  the  vanishing  terms  in  equation  (i) 
is  as  therein  written,  with  the  zero  omitted.     Thus  we  have 

3  3 

and 

•      17  =^y,\ogx  -  — +/, 

36^      9 
where 

y'  =  i§^3_47^3  +  lA^  +  ii7^+  LM:8^3  +  .  ,  1. 
9  9  3      L        4.6         4.5-6.7  J 


Imaginary  Values  of  a  and  /?. 

188.  We  have  assumed  the  roots  a  and  b  of  f{&)  =0  to  be 
real,  but  the  roots  c  and  d  of  </>({>)  ~  o  may  be  imaginary.  In 
that  case  a  and  y3  will  be  conjugate  imaginary  quantities,  say 

a  =  fjL  -\-  t'v,  13  =  fx  —  tv. 

The  integrals  will  then  take  the  form 

L  1.7  1-2.7(7  +1)  J 

and 

y^  =  ^^-vfi  +  (M  +  I  -  7);  +  ^^ 

L  1(2-7) 

+  [(/^  +  I  -  7)^  +  v^ir(M  +  2  -  y)'  -h  v^1^^,  _^         1 

1.2(2    -   y)(3    -   y)  •••J* 


206  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 88. 

Again,  when  y  is  an  integer,  making  the  same  substitutions 
in  equation  (6),  Art.  185,  the  second  integral  becomes 

where 


Infinite  Values  of  a  and  yS. 

189.  As  explained  in  Art.  165,  the  function  f{%)  must  be  of 
the  second  degree,  but  </)(^)  may  be  of  the  first  degree,  the 
equation  being  of  the  form 

(^  -  a){iy  -  b)y  -  x{&  -  c)y  =  o (i) 

The  solution  of  this  equation  is  included  in  the  general  solution 
already  given,  for  the  equation  is  the  result  of  making  d  infinite 
in 

(&  -a){&  -  b)y ^—  {&  -  e){d-  -  d)y  =  o.      .   (2) 

a  —  d 

Here  ■,  that  is,  -  takes  the  place  of  ;ir  in  the  standard 

a  —  d  y8 

form  ;  hence  equation  (5)  Art.  183,  gives  the  integral 

^Yi    ■   «/g-^    I   a(a  +  l)i8(/g  +  i)^'    ,         \ 
^\  1.7  y8  I.2.y(y+l)         (i^  ) 

for  equation  (2).  Multiplying  by  the  constant  y8^  and  then 
making  /3  infinite,  we  have  for  the  first  integral  of  equation  (i) 

y,  =  cc^U  +  ^^  +       "("+/\^  +  .  .  .) 
\  1-7  1-2.7(7  +0  / 


§  XVI.]  INFINITE    VALUES  OF  a  AND  (B.  20/ 

In  like  manner,  for  the  second  integral,  we  obtain 

;;,  =  x^^^-yF(a  +  I   -  y,   /?,   2  -  y,  ^V       y8  =  oo  . 

190.  Again,  when  <^(^)  is  a  mere  constant,  the  equation 
being  reducible  to  the  form 

{&  --  a){d- -  b)y  -  xy  =z  o, (i) 

it  is  the  result  of  making  both  c  and  d  infinite  in 

(^   _    ,)(^    _    ,)y   _    __^__(*    _    ,)(^   _    ,)y  =    o.     (,) 

We  have  now  for  the  first  integral  of  equation  (2) 

X^    f  a^  X  a(a  +   l)^(/?  +  l)     X-  \ 

a^yS'^V  i.ya/?  1.2.7(7+1)         a^/32        '")' 

Multiplying  by  af^^'^y  and  putting  a  ==  00 ,  /?  =  00 ,  the  first  in- 
tegral of  equation  (i)  is 

y,  ^x-li^-—x  + -i- — -x^  +  .  .  A 

\        i-y         1.2.7(7  +1)  / 

=  Jt«  i^/a,  /3,  7,^j,       a  =00,      y3=oo, 

and,  in  like  manner,  the  second  integral  is 

y^  =  x^^^-yF(a,(B,  2  -7.4).   a=oo,^=oo. 

If,  in  either  of  these  cases,  7  is  an  integer,  so  that  the  log- 
arithmic form  of  solution  is  required,  the  second  integral  is  given 
by  equation  (6),  Art.  185,  and  is  of  the  same  form,  except  that 
the  infinite  factors  disappear  after  multiplication  by  yS"*  or  a^^^y 
and  the  reciprocals  of  these  factors  vanish  from  the  adjuncts  in 
the  supplementary  series  (5). 


208  THE  HYPERGEOMETRIC  SERIES.  [Art.  I9I. 


Cases  in  which  a  or  ft  equals  y  or  Unity. 

191.  The  binomial  equation  of  the  first  order  may  be  reduced 
to  the  form 

(^  _  a)y  ^  x(&  -  e)y  =  o, (i) 

and,  with  the  notation  of  the  preceding  articles,  its  solution  in 
series  is 

^  =  ^(1+ ^^  +  ^i^L±Li)^.  +  ..  \ (2) 

This  is,  of  course,  the  value  of  x^(i  —xy-'*,  or  x^{i  —  ;r)-*,  which 
is  the  integral  in  its  ordinary  finite  form.  The  series  involved 
may  obviously  be  written  F(a,  y,  y,  x),  where  the  value  of  y  is 
arbitrary,  and  accordingly  this  value  of  ^  is  one  integral  of  the 
equation 

(^&-.a){&-^d)y--x{&-c){&--d-hi)y  =  o,     .     .(3) 

since  )8  =  y  in  equations  (4),  Art.  183,  makes  d  •=■  b  —  i.  The 
other  integral  of  this  equation  is 

^,  =  W.  +  -i^.+  /-^)(^--+')^  +  ...Y(4) 

\         b  —  a-\-\         {b  —  a-\-\){b  —  a-\-2)  / 

or 

y^  =  x^^-^-yPia.  +  I  -  y,  I,  2  -  y,  x), 

192,  Equation  (3)  might  have  been  solved  by  the  method  of 
Art.  141 ;  for  it  becomes  an  exact  differential  equation  when 
multiplied  by  jf-^-*  [see  equation  (i),  Art.  140].  The  result  of 
the  first  integration  is 

(^  -  a)y  -  x{&  -  c)y  =  Cx^ ; 

and  in  the  second  integration  the  value  of  7  in  equation  (2)  is 
the  complementary  function,  and  that  of  ^,  is  the  particular 


§  XVI.]     BINOMIAL   EQUATION  OF   THE    THIRD   ORDER.     209 

integral.  Thus  the  hypergeometric  series  in  which  one  of  the 
first  two  elements  is  equal  to  y  reduces  to  the  form  assumed 
when  the  equation  is  of  the  first  order,  and  that  in  which  one 
of  the  first  two  elements  is  unity  is  of  the  form  of  the  particular 
integral  of  an  equation  of  the  first  order  when  the  second  member 
is  a  power  of  x. 

The  Binomial  Equation  of  the  Third  Order. 

193.  The  binomial  equation  of  the  third  order  may  be  reduced 
to  the  form 

{p  -a)(d--  b)  {&  -  c)y  -  x(&  -d)({>-  e)  (&  -f)y  =  o. 
One  of  its  three  independent  integrals  is 

•^'        '^K^  ^.h.''^         1.2.8(8  +  i)c(.  +  I)         "+■■■}' 

where 

a^  a  ^  dj  fi  =  a  —  e,  y  —  a  — /, 

8  =  «  —  <5  4- I,  c  =  <2  —  Y"  +  I, 

and  the  other  two  are  the  result  of  interchanging  a  and  ^,  and 
a  and  c  respectively.* 

The  notation /^f*^'/^'  ^'  xA  has  been  employed  for  the  series 

involved  in  the  value  of  y^  above. 

*  When  two  of  the  roots  a,  <5,  and  c  of  /(i9)  differ  by  an  integer,  so  that  one  of 
the  quantities  d  or  £  is  an  integer,  the  powers  of  x  which  occur  in  one  of  the  three 
integrals  will  occur  in  another  with  infinite  coefficients.  By  the  process  employed  in 
Art.  185  these  infinite  terms  are  replaced  by  terms  involving  \ogx  and  the  adjuncts. 
If  both  6  and  t  are  integers,  the  third  integral  contains  terms  which  occur  in  each 
of  the  others,  with  doubly  infinite  coefficients,  and  by  a  similar  process  these  may 
be  replaced  by  terms  involving  (log;f)2as  well  as  \o%x.  Similar  results  hold  for 
binomial  equations  of  any  order.  See  American  Journal  of  Mathematics^  vol.  xi., 
PP-  49»  S0»  51' 


210  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 94. 

Development  of  the  Solution  in  Descending  Series, 

194.  When  both  of  the  functions  /  and  <^  in  the  binomial 
equation  are  of  the  second  degree,  that  is,  when  a  and  y8  are 
finite,  the  integrals  y^  and  y^  are  convergent  for  values  of  x  less 
than  unity,  and  divergent  when  x  is  greater  than  unity.  In  the 
latter  case,  convergent  series  are  obtained  by  developing  in 
descending  powers  of  Xj  or  what  is  the  same  thing,  ascending 
powers  of  x-^.     Putting,  in  equation  (i).  Art.  183, 

2  =  -,  whence  ^  =  2—  =  —&, 

X  dz 

we  have 

(^  -h  ^)(^  +  d)y  -  z{^  +a){d'  +  b)y  =  o; 

hence  the  results  are  obtained  by  changing  ^,  by  c,  and  d,  in 
the  preceding  results,  to  —Cy  —d,  —a,  and  —b.  Making  these 
changes  in  equations  (4),  and  denoting  the  new  values  of  a,  ft 
and  y  by  a,,  /?„  and  y,,  we  find 

tti  =5  —c  •\-  a  =  a, 

A  =   ~^H-^  =  a+  I  -   y, 

7i=— ^  +  ^+l=a+l--^; 

and  the  integrals  are 

P;  =  z-cF{a,,  ft,  y„  2) 

=  x'fU,    a  +  I  -  y,    a  +  I  -  ft   iV    .     .     .     .   (i) 
y.  =  z-dF{p.U  ft',  y/,  2) 

=  x^fU,  ^  +  I  -  y,  ^  +  I  _  a,  i^ (2) 


§  XVL]         TRANSFORMATION  OF  THE  EQUATION.  211 

Transformation  of  the  Equation  of  the  Hypergeometric  Series. 

195.  The  equation  of  the  hypergeometric  series, 

:.(i-:^)g+[7-^(i+a+/3)]g-a^;;  =  o,      .(i) 

admits  of  transformation  in  a  variety  of  ways  into  equations 
of  the  same  form,  leading  to  other  integrals  still  expressed  by 
means  of  hypergeometric  series.  One  such  transformation  is 
obviously  y  ■=.  x^-'*v\  for,  since 

this  will  give  an  equation  for  v  one  of  whose  integrals  is  the 
simple  hypergeometric  series  F(a',  ^,  y',  x),  that  is,  the  trans- 
formed equation  is  of  the  form  (i),  the  new  values  of  a,  ^8,  and  y 
being 

a'  =  a  H-  I  —  y, 

)8'  =  )8  +  I  -  r, 

7'  =  2  -  y. 

The  second  integral  of  this  equation  will  be 

v^  =  x^-yF{a'  +  I  -  r',  /?'  +  I  -  7',  2  -  7',  x) 

=  xy--F{a,  ^,  7,  x), 

and  the  corresponding  value  of  y  is  F{a,  /3,  7,  x),  which  is  the 
value  of  y^.  This  transformation,  therefore,  gives  no  new 
integral. 

196.  Let  us  now  make  a  transformation  of  the  form 

^  =a  (i  ~  x)f^  V. 


212  THE  HYPERGEOMETRIC  SERIES.  [Art.  1 96. 

Comparing  equation  (i)  with  the  form 


we  have 


%^'t^<^y  =  - 


x{\  —  X)  X  \   —  X 


and 


e  = 


^(i  —  a:) 


Hence,  putting  in  the  formulae  of  Art.  152  «;,  =  (i  —  xY^  we 
find,  for  the  transformed  equation, 

„  _  7  j_  7-g-^-i-2/^ 
^^  ■"  ^  "^  \  -  X 

^*         (l  -  a:)^  I  -  .^  ^ 

-_  /^(/*  -  i)  -  /^(7  -  <^  -  ^  -  i)  _  HlL±-2^, 
(i  —  xY  x{i  —  ^)* 

In  order  that  (2i  n^ay  take  the  same  form  as  2,  let  ^  be  so 
determined  that  the  first  term  of  this  expression  vanishes. 
This  gives  /a  =  o  (in  which  case  no  transformation  is  effected), 
or  else 

II  —  y  —  a  —  p. 

Then,  if  a„  /3i,  7,  have  the  same  relation  to  P^  and  (2i  that  a,  ft  7 
have  to  P  and  (2>  the  form  of  P^  shows  that  y^  =  7,  and 

tti  +  ^i  =  g  +  )3  +  2/A, 

and  that  of  Qt  shows  that 

ttxft  =  a/3  +  /X7. 


§  XVI.]         TRANSFORMATION  OF  THE  EQUATION  213 

Substituting  the  value  of  /x  above,  and  solving,  we  find 

a,  =  7  —  a, 

^x  =  7  -  A 

yi  =  7. 

The  integrals  of  the  transformed  equation  are,  therefore,. 

v^  =  Fici  -  a,  7  -  A  7.  X), 
and 

V^  =  X-'-'lFiT.   —  a,    1  —  p,    2  —  y,  x)j 

V2  being  derived  from  v^  by  the  same  changes  which  convert  y^ 
into  J2.  Denoting  the  corresponding  values  of  y  by  y^  and  y^y 
we  have  thus  four  integrals  involving  hypergeometric  series  of 
which  the  variable  element  is  x ;  namely, 

y,  =  F{a,  ft  y,  x)y 

y^  =  x^-yF{a  +  I  -  y,  y8  +  I  -  y,   2  -  y,  a:), 

y,=  (1  -  x)y  —  PF(y  -  a,  y  -  ft  y,  ^), 

y^  =  ^i-v(i  —  ^)Y-«-Pir(i  _  a,    I  —  ft   2  —  y,  ^). 

197.  The  integral  y^  is  the  product  of  two  series  involving 
powers  of  x  with  positive  integral  exponents,  each  of  which  has 
unity  for  its  first  term,  and  is  convergent  when  x  <  i.  It  follows 
that  j/3  is  a  series  of  the  same  form.  But,  from  the  process  of 
integration  in  series,  we  know  that  there  can  be  but  one  integral 
of  this  form,  namely  ji.  Hence  we  have  the  theorem,  j^i  =  y^, 
or 

F(a,  ft  y,  ^)  =   (I  -  x)y  —  PF(y  -  a,  y  -  ft  y,  x). 

In  like  manner  y2  =  J4. 


214  ^^^^  HYPERGEOMETRIC  SERIES.  [Art.  I98. 


Change  of  the  Independent  Variable. 

198.  It  is  obvious  from  the  form  of  the  equation  of  the 
hypergeometric  series,  equation  (i),  Art.  195,  that,  if  we  change 
the  independent  variable  from  x  to 

/  =  I  —  a:, 

the  first  and  third  terms  will  be  unchanged,  and  the  second  will 
be  unchanged  in  form.     The  result  is 

/(I  -  /)^  +  [i  4-  a  -f  ^  -  7  -  /(I  +  a  +  /?)]  ^  -^  a^;;  =  o  ; 

comparing  this  with  the  original  equation,  we  find  that  a  and  ^ 
are  unchanged,  while  y  is  replaced  by 

I  +  a  +  ^8  -  7. 

We  hence  derive  the  integral 

jFs  =  7^(a,  /?,  1+  a  +  y8  -  7,  I  -  X), 

A  comparison  of  this  integral  with  y^  or  F{a.,  y3,  7,  x)  shows 
that  from  any  integral  expressed  by  a  hypergeometric  series  we 
can  derive  another  integral  of  the  same  equation  by  making  the 
above-mentioned  change  in  the  third  element,  and  at  the  same 
time  changing  the  fourth  or  variable  element  to  what  may  be 
called  its  complement^  that  is,  the  result  of  subtracting  it  from 
unity.  The  process  applies  equally  well  to  an  integral  of  the 
form  y  =  wv^  where  z;  is  a  hypergeometric  series ;  for  a  new 
integral  of  the  equation  for  v  gives  a  new  integral  of  the  equation 
for  J.  Thus  the  integrals  72*  jJ^s.  and  j^  would  lead  to  the  three 
new  integrals  y^y  ye,  and  ys,  which  will  be  found  in  a  subsequent 
article. 


§  XVI.]      CHANGE   OF  THE  INDEPENDENT  VARIABLE.       21 5 

199.    It  is  shown  in  Art.  194  that,  when  we  change  the 
independent  variable  to  ^  =  -,  the  binomial  equation  retains 

its  form.  The  equation  considered  in  that  article  becomes  that 
of  the  hypergeometric  series  when  we  put  ^  =  o,  whence 
c  =  —a.     Thus  equation  (i),  Art.  194,  gives  the  integral 


Y^ 


=  X-^'Ffa,    a  +  I  —  y,    a  +  I  —  ^,  -  j 


A  comparison  of  this  integral  with  F{a,  /?,  y,  x)  gives  a 
method  by  which  we  may  pass  from  any  integral  in  the  form 
of  a  hypergeometric  series  to  another  in  which  the  variable 
element  is  replaced  by  its  reciprocal ;  and,  as  in  the  preceding 
article,  the  process  applies  also  to  an  integral  of  the  form 
J/  z=  wv,  where  ^'  is  a  hypergeometric  series. 

200.  If  we  start  with  the  variable  x,  and  alternately  take 
the  complement  and  the  reciprocal,  we  obtain  the  following  six 
values  of  the  variable, 

X    i-x       '  ^  I  -^    I 

I  —  X  1  —  X  X  X 

the  seventh  term  of  the  series  being  identical  with  the  first. 
The  corresponding  integrals  derived  from  7,  by  the  processes 
of  Arts.  198  and  199,  are 

y^   =  ^(s  A  y,        X    ), 

ys    =  ^(«»  A   I  +  a  +  )8  -  7,       1-x)  , 

=   (I  -  x)-<^F(^a,  y  -  A  r,  -73^)  > 

X-'^fL,  a+i-r,   I+a  +  i8-y,  -^-^)  » 
^-«7^^a,  a+l~y,  a+I-ft  ^     \, 


y^3 

^21    = 


2l6  THE  HYPERGEOMETRIC  SERIES.  [Art.  200. 

where,  in  writing  the  last  two,  the  constant  factor  (— i)-*  has 
been  omitted. 

201.  From  each  of  the  integrals  given  above  we  may  derive 
three  others,  exactly  as  j^z,  /s  and  y^  are  derived  from  y^.  We 
have  thus  the  following  system  of  twenty-four  integrals, 

y^  =  F{o.,  ^,  y,  X), 

y^  =  x^-yF{a  +  I  -  y,  ^  +  I  -  y,  2  -  y,  ^), 

^3  =  (i-x)y  —  ^F{y-a,  y- A  y,  x), 

y^  =  x'-y{i-x)y---PF{i-a,  I -A  2-y,  x), 

y^  =  F(a,  ft,  I  +  a  +  ^  -  y,  i-x)y 

ye  =  (i  -  x)y---PF{y  -  a,  y  -  ^  i  _  a  -  /?  +  y,  i  -  x), 

y,  =  x—yF{a  +  I  -  y,  /3  +  I  -  y,  I  +  a  +  ^  -  y,  i  -  ^), 

^/g  =  ^^-Y(l  -x)y-'^-^F{l-a,   I -13,   I  -a-/3-fy,   I  -  ^), 

j;,   =  (I  -  x)—fU  y  -  a  a  +  I  -  a  ^3^^  , 
j-.o  =   (I  -  x)-^f(b,  y  _  a,  /?  +  I  -  a,  -^^  , 

y^^  =  {-xy-y{i-x)y-^—F(i-(3,  a+i-y,  a+i-/J,  YZri)' 

y,,  =   (_:v)l-7(I_^)v-x-^7^/I_a,  )8-fi-y,  /3+i-a,  ^^]  , 

,,  =  ^i-y(l  _^)v-i-«ir/a+l-y,    1-^2-7,    -73^)' 
^^,5  =  (I  -  ^)-Pi^(y  -  a,  A  7,   -73^)  > 


§  XVI.]  THE   TWENTY-FOUR  INTEGRALS.  21/ 

jVx7  =  x-'^FU.,  a+l-y,   i+a  +  y8-y,  -^  ~  "^  j , 

;;,9  =  ^-^i^/y8  +  I  -  y,  A   i+a+/?-y,  -^^)  , 


=  ^-«/^^a,    a  +  I  —  y,   a  +  I  —  A  -J  , 
=  x-^fU,   ^  +  I  _  y,   ^  +  I  ^  a,  iV 

j3^  =  ^«-v(^  _  i)Y-«-P7^/i  -  a,  y  -    a,  y8  +  I  -  a,  -^ 


Since  the  binomial  equation  of  the  second  order  can  be 
transformed  into  the  equation  of  the  hypergeometric  series, 
it  follows  that  the  binomial  equation  has  in  general  twenty-four 
integrals  expressible  by  means  of  hypergeometric  series.*  But, 
in  the  cases  considered  in  Arts.  189  and  190,  where  a  or  /?  is 
infinite,  we  have  only  the  integrals  y^  and  y^. 


*  The  twenty-four  integrals  are  written  above  exactly  as  they  arise  in  the 
process  indicated,  except  that  the  factor  ( —  \Y~'^  is  dropped  in  the  case  of  J14 
and  /i6,  and  (_i)Y  — *-^  is  dropped  in  the  case  of  jj/is  and  jao-  Because  y-i  =  js 
and  y^  ■=.  y^,  the  first  and  third  integral  of  each  group  are  equal,  and  so  also 
are  the  second  and  fourth,  the  omission  of  a  factor  in  the  cases  mentioned  above 
causing  no  exception.  It  may  also  be  shown,  by  comparing  the  developments  in 
powers  of  x,  that  the  integrals  of  the  first  group  are  respectively  equal  to  those 
of  the  fourth  group,  and  those  of  the  second  to  those  of  the  fifth  group.  But  in 
the  third  and  sixth  groups  jg  =  {— i)";y2i  and  jio  =  (  —  1)^^22-  Thus  the  twenty-four 
integrals  consist  of  six  sets  of  equal  quantities,  as  follows :  — 


2l8  THE  HYPERGEOMETRIC  SERIES.  [Art.  202. 


Solutions  in  Finite  Form. 

202.  The  condition  that  jF(a,  /?,  y,  x)  may  represent  a  finite 
series  is  readily  seen  to  be  that  one  of  the  elements  a  or  /8  shall 
be  zero  or  a  negative  integer.  But,  since  y^  =  73,  the  form  of 
J3  shows  that,  if  either  y  —  a  or  y  —  y3  is  zero  *  or  a  negative 
integer,  /^(a,  y8,  y,  ;r)  may  be  expressed  in  finite  algebraic  form. 
For  example,  one  integral  of  the  equation 

2x{\--x)^-\-{\  —  {2n-\-  <,)x\-J-- iny  ^  o 

is  the  infinite  series  represented  by  i^(|,  «,  J,  x).  Here  y  --  a  is 
a  negative  integer,  and,  using  the  form  y^  the  integral  may  be 
written 

(i  _  x)-^--F{-i,   i  -  n,  i,  x), 


yi  =  yi  =  yi3  =  yis, 

j^2  =  y4  =  yu  =  yi6j 

ys  =  y?  =  yn  =  yi^^ 

ye  =  yt  =  yis  =  J20, 

^10  =  J12  =  (-i)^>'22  =  i-ifyu- 

Between  any  three  integrals  belonging  to  different  sets  there  must  exist  a 
relation  of  the  form  y^  =  Afy^  +  JVjy^.  These  relations,  in  which  the  values  of 
M  and  iV  involve  Gamma  Functions,  are  equivalent  to  those  given  by  Gauss  in  the 
memoir  "  Determinatio  Seriei  Nostrae  per  Aequationem  Differentialem  Secundi 
Ordinis,"  IVerke,  vol.  iii.  See  equations  [86],  p.  213,  and  [93],  p.  220.  The 
twenty-four  integrals,  and  their  separation  into  six  sets  of  equal  quantities,  were 
first  given  by  Kummer,  in  a  memoir  "  Ueber  die  hypergeometrische  Reihe,"  Crelle^ 
vol.  XV.,  p.  52.  The  order  of  the  integrals  is  different  from  that  given  above,  and 
some  errors  involving  factors  of  the  form  (—1)'^  occur  in  the  statement  of  the 
equalities.  The  values  of  M  and  N  are  given  by  Kummer  for  the  integrals 
numbered  by  him  i,  3,  5,  7,  13,  and  14,  corresponding  to  the  integrals  71,  j2> 
yi^  y6f  ^9.  and  >'io  above. 

*  The  case  in  which  y  —  0  =  o  has  already  been  considered  in  Art.  191. 


§  XVL]  SOLUTIONS  IN  FINITE  FORM.  219 

in  which  the  second  factor  is  the  finite  series 

I  H ^'^—^ -X  =  I  +  \2n  —  i)^. 

Hence  the  integral  in  question  is 

(i  —  x)»  +  ^ 

203.  In  like  manner  the  integral  ^2  will  be  a  finite  series  if 
either  of  the  quantities  a+i—yoryS+i  —  yis  zero  or  a 
negative  integer ;  and,  since  ^2  =  J4>  the  form  of  j/^  shows  that 
if  either  i  —  a  or  i  —  /?  is  zero  or  a  negative  integer  (in  other 
words,  if  a  or  y8  is  a  positive  integer),  j/2  may  be  expressed  in 
finite  form.     It  will  be  noticed  that  the  eight  quantities, 

«;   /3,  y  —  a,  y  —  /3,  a  +  I  —  7,  ^  +  I  —  y,    I  —  a,    1  —  jS, 

are  the  only  values  of  the  first  two  elements  in  the  twenty-four 
integrals  ;  hence  the  only  cases  in  which  they  furnish  finite  in- 
tegrals are  those  in  which  either  a,  /?,  y  —  a,  or  y  —  ^  is  an 
integer. 

In  the  case  of  the  general  binomial  equation  of  the  second 
order,  the  condition  given  in  the  preceding  article,  when 
applied  to  both  integrals,  is  sufficient  to  determine  whether 
finite  algebraic  solutions  exist.* 


*  Finite  solutions  involving  transcendental  functions  occur  in  certain  cases 
considered  in  the  following  chapter.    See  Arts.  209,  213,  214,  and  217. 


220  THE  HYPERGEOMETRIC  SERIES.  [Art.  203. 

Examples    XVI. 

1.  Show  that,  in  the  notation  of  the  hypergeometric  series, 
(/  +  uy  +  (/  -  uy  =  2tnF[-\n,  -\n  +  J,  \,  ^^ , 

(/  +  uy  -  {t  -  uy  =  2ntn-^uFl-\n  +  J,  - 1«  +  i,  f,  ^^ , 
\og{i  -^  x)  =  xF{iy    I,    2,    -a:), 
logi-^  =  2XF{\,    I,   f,   A-), 

f-=  i^^i,  >^,   I,  1^  =  I  +  xfI^i,  k,  2,  |^ 

=  I  +  a:  +  ^A:*i^(  I,  ^,  3,  -J  =  etc.,  where  ^  =  00, 

sin^  =  xF\k,  k\  f, —\  ,  k  =  k'  =  c^, 

cos:v  =  7^^^,   /&;   i,    -  -^^ ,  k  =  U=:z^, 

cosh^  =  j^^^,  k',  i,  -^V  k=.k'=^, 

sin-^AT  =  xF{\,  J,   f,  ^2), 
tan-^:x:  =  xF(^\,    I,   f,    -x^), 

2.  Show  that 

£7^(a,ft  y,  ^)  =  ^ir(a  +  i,  ^  +  i,  y  +  i,  ^), 

£F(a,P,y,x)  =  ^(l±^M^i.(a  +  2,/3  +  2,y  +  2,.),etc. 
OV*  y(y  +  i) 


§  XVL]  EXAMPLES.  221 

3.   Show  that  the  equation 

Ay  Ji^iB  -^  Cx)^  -\-{D  +Ex  +  Fx')p^  =  o 
ax  do^ 

can  be  reduced  to  the  equation  of  the  hypergeometric  series,  and  hence 
that  the  complete  integral  is 

A 

where  a  and  b  are  the  roots   oi  D  -\-  Ex  -f-  Fx^  =  o,  a)3  =  — , 

F 

a  +  ;8  +  I   =  f ,    y  =   f  +  ^^^    and  y' =    ;|-^^  ,  the  two 

7^  (^  —  b)F  {b  —  d)F 

independent  integrals  being  related  as  y^  is  to  y^  in  Art.  198. 


4.  Find  the  particular  integral  of  the  equation 

(^  -  a){^  -  b)y  -  x{&  -  c){&  -  d)y  =  kxP, 
and  derive  the  integrals  in  Art.  183  from  the  result. 

Solve  the  following  equations :  — 

5.  x(i  -  x)^  +  (I  -  2^)g  -~iy  =  o, 

y  =  AF{h,  h  i,  X)  +  Bx-\  =  -^^'"-)g  +  ^ . 

\x 


6.    2J£:(l   —  x)-^  +  x-/  —  y  =  o, 
dx^  dx 

y^x{A-\-B\ogx)-\-Bl2  +  ^-x^  +  ^"^  +  h^J^  4- .  . . V 
\        4  4-62       4.6.8  3  / 


232  THE  HYPERGEOMETRIC  SERIES.  [Art.  203. 

7.  Transform  the  series 

,      8       ,      8.10  -    ,      8. 10.12   ,   , 

^  =  I  +  2-x  +  3 ^  +  4 ^^  +  . . . 

9  9.11  9-II-I3 

by  means  of  the  theorem  of  Art.  197. 

Solve,  in  finite  form,  the  following  equations  :  — 


-)-5#,+(- 

=  0, 

y^ 

^  I  +  6*  +  *» 

+  B 

(I- 

-xY 

-)S+*('- 

-  2^)|  +  ¥^  = 

--  0, 

9.  ;r(i 

ax- 

y  =  A(i-'  x)^(i  -  6x)  H-  Bx^S  -  6a:). 

10.  2:c(i  -  a:)^  4.  ^  +  4^  =  o, 

^::c*         ax 

y  =x  ^(3  -  i2a:  +  8a^)  +  ^A;i(i  -  x)^, 

11.  Solve  the  equation 

first  transforming  to  the  new  independent  variable  z  =  1  —  ^. 

y  =  A{i  -  x^)i  +  ^^(i  -  x^)^. 

12.  When  a  is  a  negative  integer,  the  six  integrals  of  Art.  200  are 
all  finite  series,  and  therefore  must,  in  that  case,  be  all  multiples  of  the 
integrally,.     Verify  this  when  a  =  —i. 

13.  Show,  by  comparing  the  first  two  terms  of  the  development, 
that  7,  =  ^,3,  and  thence  that 

F{a,  ft  y,  sin*  6)  =  (cos*  e)y---^F{y  -  a,  y  -  ft  7,  sin*  6) 
=  (sec*  6YF{a,  y  -  ft  y,  -tan*  6) 
=  {sQc^eyFly  -  a,  ft  y,  -tan*^). 


§  XVI.]  EXAMPLES.  223 

14.  From  the  expression  for  sin-'^  as  a  hypergeometric  series, 
derive 

^  =  sin^/^(iif,  sin^^) 

=  sin^cos^i^(i,  I,  I,  sin*^) 
=  \xciBF{\,  i,f, -tan'^). 

15.  The  integrals  of  the  equation 

are  sin  nB  and  cos  nB ;  form  the  equation  in  which  ^  =  sin  ^  is  th6 
independent  variable,  and  thence  derive  four  expressions,  as  in  Ex.  13, 
for  each  of  these  quantities. 

sin«^  =  n%mBFi^\  -  \n,\-\-  \n,  f,  sin*^), 

=  ;2sin^cos^i^(i  ■\-  \n^  \  —  \n,  f,  sin^^), 

=  «sin^(cos^)«-^i^(i  -\n,\-  \n,  |,  -tan*^), 

=  /2sin^(cos^)-«-^i^(i  +  |«,  J  +  \n,  f,  -tan*^); 

cos«^  =  F{—\n,  ^n,  ^,  sin*^), 

=  cos  BFii  +  ^n,^  -  in,  J,  sin*  6), 

.     .       =.  (cos  ByFi-^n,  J  -  i«,  i,  -tan*  B), 
=  (cos  B)—F(in,  i  +  ^n,  |,  -tan*0). 

1 6.  Denoting  by  i?  the  expression 

^(^  -  i)^  +  (3^  -  i)f  +  x^ 
dx^  ax 

show  that  the  equation  xt—  +  (i-\-  xx-f\R  =  o  is  equivalent  to 
dx        \  dxj 

^(^  -  i)^  +  3^(3^  -  i)^  +  (19^  -  i)^  +  8^'«  =  o, 

</jt:3  ^/*  dx 

where  u  ^  P ;  and  thence  that 

23     ^  23.43       ^  \  4*  4*.8*  / 

^  Gauss,  WerkCy  vol.  iii.  p.  424. 


224  Rice  ATP  S  EQUATION.  [Alt.  204. 


CHAPTER   IX. 

SPECIAL   FORMS   OF    DIFFERENTIAL   EQUATIONS. 

XVII. 
Riccati^s  Equation. 

204.  There  are  certain  forms  of  differential  equations  which, 
either  for  their  historic  interest  or  their  importance  in  mathe- 
matical physics,  deserve  special  consideration.  Of  these  we 
shall  consider  first  Riccati's  equation  and  its  transformations. 

The  equation 

^  +  ^^a  =  cx^ (i) 

dx 

was  first  discussed  by  Riccati,  and  attracted  attention  from  the 
fact  that  it  was  shown  to  be  integrable  in  a  finite  form  for 

certain  values  of  m.     If  we  put  -  in  place  of  Xy  and  write  a"*  for 

0 

the  constant   -^ — ^  the  equation  becomes 

tfnt  +  i 

^+y^  =  a'x^, (2) 

ax 

so  that  no  generality  is  lost  by  assuming  the  coefficient  5  equal 
to  unity.  The  case  in  which  the  coefficient  of  x*"  is  negative 
will  be  provided  for  by  changing  a^  to  —  ^%  that  is,  a  to  ia,  in 
the  results. 


§  XVII.]  STANDARD  LINEAR  FORM.  22$ 

205.  In  the  form  (2),  Riccati's  equation  is  the  equation  of 
the  first  order  connected,  as  in  Art.  151,  with  the  Hnear  equation 
of  the  second  order, 

a^x^'u^o;      .     .     .     .     .     .     .   (3) 

in  other  words,  this  last  equation  is  the  result  of  the  substitu- 
tion 

__\  du 
udx 

in  equation  (2) ;  and,  denoting  its  complete  integral  by 

u  =  c^X^  +  c^X^,    ........  (4) 

that  of  equation  (2)  is 

c^Xl±c^  ^Xl±jcXl  .. 

^      c,X,+c,X,       X^^cX,'     •    •    •    •    •  V5; 

which  shows  the  manner  in  which  the  constant  of  integration 
enters  the  solution. 

Standard  Linear  Form  of  the  Equation. 

206.  The  discussion  of  Riccati's  equation  is  simplified  by- 
using  the  linear  form  (3)  ;  moreover,  the  expression  of  the 
results  and  transformation  to  other  important  forms  is  facilitated 
by  writing  the  exponent  m  in  the  form  2q  —  2.'^  We  shall, 
therefore,  take 

d'^u  r  X 
a'^x^9-^u  =  0 (I) 

dx^  ^ 

as  the  standard  form  of  Riccati's  equation  from  which  to  deter- 

*  This  improvement  of  the  notation  was  introduced  by  Cayley,  Philosophical 
Magazine,  fourth  series,  vol.  xxxvi.,  p.  348. 


226  RICCATPS  EQUATION.  [Art.  206. 

mine  the  independent  integrals  X^  and  X^ ;  the  integral  of  the 
equation  in  the  original  form  being  then  given  by  equation  (5) 
of  the  preceding  article. 

Substituting  in  equation  (i) 

we  have 

2*[(;w  ^2qr){m  +  2qr  —  i)^^a:'«  +  «^'— =» —^•^^a:'« +  »«"'+ ^^-»]  =  O. 
Equating  to  zero  the  coefficient  of  ji:'«  +  2yr-2^  ^g  have 
{tn  +  2qr)  {m  +  2^r  —  i)^r  =  a^Ar-i, 

and,  when  r  =  o, 

m{m  —  i)Aa  =  o, 

whence  m  =  o  or  m=  i.  Taking  w  =  o,  we  obtain  the  in- 
tegral 

«x  =  I  +  — --^— -^«^+  ; '^\.      •  ,^4^  +  .  .  ., 

2q{2q-i)  2q.4q(2q-  i){4q--i) 

and,  taking  w  =  i, 


U2  =  xfi  -\ r-^^-: — r^^H ; r^^-rr ^ — r^'^^+  •  •  •!• 


2^(2^+1)         '     2q,4q{2q  +  i)  {4q -{- i)' 


207.  The  integrals  u^  and  «»  are  in  no  case  finite  series,  nor 
do  they  fulfil  the  condition  given  in  Art.  202  for  expression  in 
finite  form,  since  in  the  notation  there  employed  u  and  /3  are 
infinite.     Let  us,  however,  apply  the  transformation, 

u  =  e^'^v, 

considered  in  Art.  154,  and,  if  possible,  determine  a  and  m  in 
such  a  manner  that  the  transformed  equation  shall  still  be 
binomial.     The  equation  for  v  is 


§  XVII.]  INTEGRALS  IN  SERIES.  '227 

dx^  dx 

which,  it  will  be  noticed,  becomes  a  binomial  equation  if  we  put 

m  =  q  and  m^'a'^  =  a^,  whence  a  =  ±  -.     Hence  we  may  put 

a    q 

U  =  e'^     V, 

the  transformed  equation  then  being 

^-\.2ax^-''^-\-a{q--i)x^-^v^O',    .     .     .     .   (2) 
dx^  dx 

and  in  the  results  we  may  change  the  sign  of  a,  as  is  indeed 
evident  from  the  form  of  equation  (i). 

208.    Putting  in  equation  (2)  v  =  S^^r^'"'^''^*  we  have 

^rCC^  +  r^)(w  +  rq  ^  i)ArX'"  +  *'^-'' 

+  2a(m  4-  rq)ArX»*  +  *'^  +  f-''  +  a{q  ~  i)^^x'«  +  ''^ +  «'-»]  s=  o; 

whence,  equating  to  zero  the  coefficient  of  ;r'«  +  '*?-2,  we  derive 

(m  -\-  rq)  {m  -\-  rq  ^  i)Ar  +  a{2m  •\-  2rq  —  q  —  i)Ar-x  =  O, 

and,  when  r  =  o,        - 

m{m  —  \)Ao  —  o, 

whence  m  =  o  or  m=  i.     Taking  w  =  o,  we  have 

A    =-  (2r-  l)^-  I 
rqirq-  i) 

which  gives,  for  the  first  integral  of  equation  (2), 


z/,  e=  I  — 


?(?-0  ?-2?(?-  l)(2?-  I) 


228  RICCATrS  EQUATION.  [Art.  208. 

and,  taking  w  =  i, 

rq{rq-\-\) 
which  gives,  for  the  second  integral, 

We  have  thus  two  new  integrals  of  the  equation 

'±-^  —  a^x^9-'u=z  o, 
namely, 

=  e^V,  _  -J-^ax^  +      (^-i)(3^-i)       .^^  ^  .  .  V 

\      $^(^-0         ^.2^($'- 1)(2^- 1)  y 

I 

\     ^(^  +  i)       ^.2^(^  + i)(2^+i)  y 

"   Again,  changing  the  sign  of  a,  we  have  two  other  integrals, 
namely, 

u^  =  ,-rV,  +  -f^±-ax^  +      (^-i)(3^-i)       ,^^  ^  .  .  1 

\  $^(^-1)  ^•2^(^-  l)(2^-  I)  J 

and 
«,  =  ^r^Vx  +  -f^.ax^  +      ^^+;H3^+0       .^.,  +  .  .  .\ 

\  ^(^+l)  $'.2^(^+  l)(2$^+l)  y 


i^r«/V!f  Solutions, 

209.   When  ^  is  the  reciprocal  of  the  positive  odd  integer 
2/&  —  I,  a  zero  factor  occurs  in  the  numerator  of  the  coefficient 


«3 

and 
u 


§  XVII.]  FINITE  SOLUTIONS.  229 

of  x^^  and  of  every  subsequent  term  in  the  series  contained  in 
u^.  Notwithstanding  the  fact  that  the  same  zero  factor  occurs 
in  the  denominator  of  the  same  or  a  subsequent  term,  we  have 
then  a  finite  solution,  as  explained  in  the  foot-note  on  page  175. 
At  the  same  time,  m^  gives  a  finite  integral,  which  is  simply  the 
result  of  changing  the  sign  of  a  in  that  derived  from  %  For 
example,  if  ^  =  i,  ti^  gives  the  integral  e^^y  and  u^  gives  e^'^^y 
so  that  we  may  write  the  complete  integral  u  =  Ae^"^  -f  Be-'^-^. 
Again,  when  ^  =  J,  the  equation  in  the  linear  form  being 

a^x    3^  =  0, 

the  two  finite  integrals  are 

^3«*  (i  —  .3^^^),         and        ^-3«^'(i  +  3«;e^). 

Since  m  =  2g  —  2,  the  equation  of  Riccati  in  its  non-linear 
form  (2),  Art.  204,  is  in  this  case 

^-^y^=za^x-i,  ; 

ax 

and,  substituting  in  equation  (5),  Art.  205,  the  results  just  found, 
the  complete  integral  of  this  equation  is  found  to  be 


(i  -  2>ax^)e^^^^  +  c{i  +  3«jc*)^-3«^' 

We  may,  if  we  please,  express  this  solution  in  a  logarithmic 
form  ;  for,  solving  for  Cy  we  have 

^  T  ■  —  ^.»    , 


230  RICCATPS  EQUATION.  [Art.  209. 

whence 

lax^y  +  x^y  +  3^* 

210.  In  like  manner,  if  q  is  the  reciprocal  of  a  negative  odd 
integer,  u^  and  //6  give  finite  independent  integrals.  Hence  we 
have  a  complete  solution  in  finite  terms,  when  q  is  of  the  form 

—: ,  where  k  is  any  integer.     Substituting  this  expression 

in  w  =  2^  —  2,  we  have 

2  —  4^ 


2^+1  2/^+1 

where  ^  is  any  integer.  Changing  the  sign  of  k,  this  expression 
becomes  —r^ —  ;  so  that  the  condition  of  integrability  in  finite 
form  for  equation  (2),  Art.  204,  is  that  m  should  be  of  the  form 

2k  ±  i' 
where  k  is  zero  or  a  positive  integer. 


Relations  between  the  Six  Integrals. 

211.  Since  ?^3,  Art.  208,  is  an  integral  of  equation  (i),  it 
must  be  of  the  form  Au^  -\-  Bu^,  where  Uj  and  U2  are  the  inte- 
grals given  in  Art.  206.     Now 

^  q     ^  ^«  2  !  ^  ^3  3  ! 

and  «3  is  the  product  of  this  series  and  the  series 

a  g   ,    a^  -ig  —  1  x^^ 
q  q^  2q  —  1  2  \ 


§  XVII.]        RELATIONS  BETWEEN  THE  SIX  INTEGRALS.       23 1 

This  product  is  a  series  having  for  its  first  term  unity,  and 
proceeding  by  integral  powers  of  x^.  But  u^  is  a  series  involving 
these  powers,  while  in  general  tt^  contains  none  of  these  powers. 
It  follows  that,  putting  ti^  =  Ati^^  +  Bti^,  we  must  have  B  =  o  and 
A  =  I  ;  that  is,  u^  =  ti^,  the  odd  powers  of  x^  vanishing  in  the 
product.     In  like  manner  we  can  show  that  u^  =  ti^,  and  that 

?/6  =  2^4  =  t^2' 

212.  It  thus  appears  that  u^  and  u^  are  not  independent 
integrals,  but  merely  different  expressions  for  the  same  func- 
tion ;  nevertheless  we  have,  in  Art.  209,  derived  from  them 
independent  integrals  in  the  case  where  they  furnish  finite 
expressions,  namely,  when  ^  is  the  reciprocal  of  a  positive  odd 
integer.  The  explanation  is  that  the  finite  expressions  are  nof 
the  actual  values  of  71^  and  n^,  in  these  cases,  but  the  results  of 
rejecting  from  the  series  involved  the  infinite  series  of  terms  in 
which  the  vanishing  factor  occurs  in  the  denominator  as  well  as 
in  the  numerator  of  each  coefficient.  The  rejected  part  of  the 
series  Vj_,  Art.  208,  is  a  multiple  of  the  series  v^ ;  so  that  the 
finite  expression,  which  we  may  denote  by  [u^],  differs  from  ii^ 
by  a  multiple  of  u^. 

The  expansion  of  the  complete  product  u^  is  still  the  series 
//i,  consisting  of  even  powers  only  of  x^ ;  but  that  of  the  product 
[^3]  contains  also  odd  powers  of  x^.  These  odd  powers  are 
accompanied  by  odd  powers  of  a ;  hence,  since  [u^]  is  the  result 
of  changing  the  sign  of  a  in  [u^],  it  is  evident  that  we  shall 
have 

For  example,  when  ^  =  i,  [u^]  =  ^•^,  and  [u^]  =  ^-'^•^,  of  which 
the  expansions  contain  both  even  and  odd  powers  of  ;ir,  but  Uj,  is 
the  even  function  cosher  =  ^(e*^^  +  ^-''^). 

In  like  manner,  when  g  is  the  reciprocal  of  a  negative  odd 
integer,  we  have  the  independent  integrals  [u^]  and  [ue],  and 

«2  =  iM+iM. 


232  TRANSFORMATIONS   OF  [Art.  2 1 3. 


Transformations  of  RiccaiiU  Equation. 

213.    Certain  important  differential  equations  may  be  derived 
by  transforming  Riccati's  equation, 

—  —  a^x'^^-^u  =  o (i) 

dx-^ 

For  this  purpose  it  is  convenient  to  use  the  t?-form  of  the  equa- 
tion, namely, 

{}{n  —  i)u  —  a^x'^^u  =  o (2) 

Let  us  first  change  the  independent  variable  from  x  to  ^, 

where 

z  =  mx^^  whence  t>'  =  2—  =  -»9. 

dz      q 

The  result  is 

tn'^ 

putting  m  =  -J  and  writing  »?  and  x  in  place  of  ^?'  and  Zy  this 

becomes 

^'^{f^  —  m)u  —  a^x^u=^  o, (3) 

which  in  the  ordinary  notation  is 

d^u  _  in  —  I  ^  _    2    _  /  \ 

dx"^  X      dx 

I  X 

Hence,  putting  ^  =  — ,  and  writing  —  in  place  of  x^  in  the 
in  nt 

six  values  of  ti  given  in  Arts.  206  and  208,  we  have  the  follow- 
ing six  integrals  of  equation  (4), 

a^      x^   ,  a*  X* 

m  —  2  2        {m  —  2)  {m  —  4)  2^2  ! 


\         m  -\t  2   2        (w  4-  2)  (/«  4-  4)  2*2  !  J 


§  XVIL]  RICCATPS  EQUATION.  233 

«3  =  e'^^i  I ax  +  ) (^^ ^  — -  -  .  .  .   , 

\         m  —  1  {m—i){m  —  2)    2\  J 

2^.  =  e^^x*"  [  I ' — ax  +  -^ ■ — f-^ !— ^ —  ...  I, 

*  \        m  +  i  {m  +  1)  {m -\- 2)    2!  y 

^■r-f      ,   m  —  1        ,    (m  —  i){m  —  -i)  a^x^    ,  \ 

\         m—i  {m—i){m  —  2)2\  J 

U(,  =  e-'^^x^i  I  H ! — ax  +  -'^ ' — '— -^^ 1-  .  .  . ). 

\         m -\- 1  {m  +  1)  {m -^  2)    2  \  J 

The  factor  m^  has  been  omitted  in  writing  u^,  u^,  and  ue,  but 
we  still  have  ti^,  =  u^  =  u^,  and  ti^  =  u^  —  U(,. 

Equation  (4)  is  integrable  in  finite  terms  when  tn  is  an  odd 
integer,  the  complete  integral  being  A\ti^  ■{■  B\u^  when  m  is. 
positive,  and  A  [?/ J  +  B\ti^  when  m  is  negative. 

214.  If  in  equation  (3)  v/e  put  m  z=z  2p  -\-  i,  and  make  the 
transformation 

u  =  x^v^ 

we  have,  since  y'^x^  V=px^  V+x^f^  V=  x^{n  +/>)V, 

('"^  +/)  ('^  — /  —  0^  —  ^''^^^  =  05 

which  in  the  ordinary  notation  is 

^2^  p(p  4.  i) 

dx^  x^ 

'  This  equation  is  integrable  in  finite  terms  when  /  is  an 
integer.*  The  case  in  which  p=2  occurs  in  investigations 
concerning  the  figure  of  the  earth. 


*  See  the  memoir  "  On  RiccATi's  Equation  and  its  Transformations,  and  on  some 
Definite  Integrals  which  satisfy  them,"  by  J.  W.  L.  Glaisher,  Philosophical  Transac- 
tions for  1881,  in  which  the  six  integrals  of  this  equation  are  dediiced  directly,  and 
those  of  the  equations  treated  in  the  preceding  articles  are  derived  from  them. 


234  BESSEVS  EQUATION.  [Art.  2 1 5. 


BesseVs  Equation, 

215.  If,  in  equation  (3),  Art.  213,  we  put  w  =  2«  and 
^'  =  —  I,  and  make  the  transformation  u  —x*^y,  the  result  is 

(,9»  _  «.)^  +  ^2j,  =  o, (i) 

or,  in  the  ordinary  notation, 

which  is  known  as  BesseVs  Equation.  Making  the  substitutions 
in  the  values  of  u^  and  /^j,  Art.  213,  and  denoting  the  corre- 
sponding integrals  of  Bessel's  equation  by^^  and  _;/_„,  we  have 

«  /  ^      x'^  ,  I  ^4  \ 

•^''  V  «  +  I    2»  («+  l)(«  +  2)    24.2  !  "  */ 

y-n  =  x-^[\  + -+ — -+  .... 

\  «  —  I    2'  («—  l)(«—  2)    24.2!  J 

It  will  be  noticed  that  either  of  these  integrals  may  be 
obtained  from  the  other  by  changing  the  sign  of  n,  which  we 
are  at  liberty  to  do  by  virtue  of  the  form  of  the  differential 
equation. 

216.  The  integrals  corresponding  to  the  other  four  values  of 
u  in  Art.  213  are  imaginary  in  form.  Making  the  substitutions 
in  the  value  of  w^,  we  may  write,  since  //^  =  ti^.  =  ^O'w* 

yn  —  x»  (cos  X  -\-i  sin  x)  {Pn  —  iQn) » 
in  which 

P  -  T  _  (2^4-  i)(2;z  +  3)  x^   . 
(2«  +  l)(2«  -h  2)  2 ! 

Q^^2n-\-i^       {2n-^i){2n  +  7,){2n  +  ^)x^    ^ 

2«H-I  (2«  +  l)(2«  +  2)(2«  4-  3)  3  ! 


§  XVI I.]  FINITE  SOLUTIONS.  2$$ 

The  value  of  f„  derived  from  ue  is  the  same  thing  with  the 
sign  of  i  changed ;  hence  we  infer  that 

y„  =  x»  {Fn  cos  X  +  Q„  sin  x) , 
and  also  that 

Pn  sin  X  —  Qn  cos  X  =  o* 

Changing  the  sign  of  n,  the  other  integral  of  Bessel's  equa- 
tion may,  in  Hke  manner,  be  written  in  the  form 

y-n=  X-^'iP-n  cos  X+  Q-nSiux), 

where 

{2n  —  i)  yzn  —  2)  2  ! 
^      _  2n-\  {2n-  i){2n-z){2n-s)  x^ 

2«  —  I  (2«  —  l)(2«  —  2)(2«  —  3)    3  ! 


i7«/V<f  Solutions. 

217.  The  case  in  which  Bessel's  equation  admits  of  finite 
solution  is  that  in  which  n  is  one-half  of  an  odd  integer.  Taking 
n  to  be  positive,  the  series  P_„  and  (2_«  contain,  in  this  case, 
terms  whose  coefficients  have  zero  factors  in  the  numerators. 
Denoting  by  [/^-«]  and  [2_«]  the  finite  series  preceding  these 
terms,  we  have,  as  explained  in  Art.  212,  an  integral  [j_„]  in 
finite  form,  but  differing  in  value  from  j_„.     Thus 

*  The  resulting  value  of  tan  x  may  be  written 

^    ^  +  5  -y^   J   (w  +  7)  (^  +  9)  ^^  • 

w  +  2  3 !       (w  +  2)  (w  +  4)  5 ! 
tan  X  = 


_  ^  +  3  £!  ^  ("^  +  5)  (^^  +  7)  £l 
m  +  2  2l       (w  +  2) (w  +  4)  4 ! 


in  which  f«  may  have  any  value. 


236  BESSEVS  EQUATION.  [Art.  21/. 

[^-«]  =  ^-"(€08^  +  ismx)\lP-n']  -  /[<2-«]| 
=  ^-"jcos;*;  [/L„]  4-sinjc[^_„]J 

-\-  ix-»\^inx\_F-„']-cosx\_Q-„']\, 

in  which  the  coefficient  of  i  does  not  vanish,  as  it  does  in  Art.  216. 
If  we  substitute  this  expression  in  the  differential  equation,  it 
is  evident  that  the  real  and  imaginary  parts  of  the  result  must 
separately  vanish,  so  that  we  have  the  two  real  integrals 

^,  =  ^-«{cosa:[/!_„]  4- sinx[^_JJ, 
and 

1/2=  ^-«Jsinjc  [Z'-m]  —  cos^[^_„]j. 

The  complete  integral  may  therefore,  in  this  case,  be  written 

y  =  Cx-*^\lF-n'\  cos  {x^a)  +  [(2-„]  sin {x  +  a)\, 

where  C  and  a  are  the  constants  of  integration. 

218.  Comparing  the  integrals  t/i  and  773  with  j_«  and  y^ 
Art.  215,  it  is  evident  that,  since  cos ;r  [/*_„]  +  sin;tr[2-«]  is  an 
even  function,  and  sin:r[P_„]  —  cos^[(2-«]  is  an  odd  function, 
the  development  of  771  contains  only  the  powers  of  x  which  occur 
m  y-„,  and  r/a  only  those  which  occur  iny„.  Moreover,  the  first 
coefficient  in  -q^  is  unity.  It  follows  that  r/i  =  j-«,  and  that 
Va  is  the  product  of  j„  by  a  constant.* 


*  To  find  this  constant,  we  notice  that  the  part  of  the  series  /*_„  —  iQ-ny  which 
is  rejected  from  the  value  of  >'_„,  when  we  use  the  finite  expressions,  as  in  Art.  217, 
commences  with  the  term  containing  x'^*^.  Denoting  the  coefficient  of  this  term  by 
A,  the  rejected  part  oiy^^t  is  Ay„.    Thus 

y-„=  [  J_  „]  +  Ay^  =  77,  +  ^■7?2  +  Ay„. 

A 
But  we  have  shown  that  tjj  =>'_„;  hence  7/3  =  — -y^,  where  A  is  the  coefficient 

of  x^*^  in  P—n  —  iQ—ny  ^^^*  ^^*  ^^  "~  ^Q—n'  "^^^  ^^^>  since  2w  is  an  odd  integer. 
Thus 

(2»-l)!(2«)!       '* 


§  XVII.]  THE  BESSELIAN  FUNCTION.  23/ 


The  Besselian  Function. 

219.    If,  when  ;^  is  a  positive  integer,  we  multiply  j„,  Art. 
215,   by  the  constant  — L_,  the  resulting  integral  of  Bessel's 

2«/2  ! 

equation  is  known  as  the  Besselian  function  of  the  n\h  order, 
and  is  denoted  byy«.     Thus 

T  x^    f  I       x^    ,  I  :r4 


2"n  \\         «  +  I  22       (^n  -\-  i)  {n  +  2)  242  ! 

=  2°°      (-  O"*     /^f Y "*"  ^*' 
°  {n  +  r)\r\\2j 

More  generally,  for  all  values  of  n  we  may  write 

/  —  ^"         /    _      I      ^  J I x^  __        \ 

~"  2«r(«  H-  l)  \  «  +  I   22         («  +  l)  («  +  2)    242  !         '  '  'J 

and  then,  in  general,  the  complete  integral  of  Bessel's  equation 
is 

where  y_„  is  the  same  function  of  —  n  thaty„  is  of  n.  It  is 
to  be  noticed,  however,  that  the  factor  which  converts  the 
series  j/_„  to  y_„  is  zero  in  value  when  «  is  a  positive  integer. 


Substituting  the  values  of  rj^  and  }f„,  Arts.  217  and  215,  we  have,  for  the  devel- 
opment of  the  odd  function  smxlP_„']  —  cos;r[^_„], 

(2«  —  l)!  (2»)!  \  2(2«  +  2)        2.4(2«  +  2)(2«  +  4)  / 


238  BESSEVS  EQUATION.  [Art.  2 1 9. 

The  series  in  this  case  contains  infinite  terms  which  are  thus 
rendered  finite,  while  the  finite  terms  preceding  that  which 
contains  x^*"  are  made  to  vanish.  The  result  is  that,  when  n  is 
an  integer, 

/_„  =(-!)«/«, 

and  the  expression  AJ„  +  BJ-n  fails  to  represent  the  complete 
integral.  The  second  integral  in  this  case  takes  the  logarithmic 
form,  and  is  found  in  Art.  221. 

220.   The  expression  forj^„,  given  in  Art.  216,  shows  that 


where 


■^-  =  ?axfTT)(^"'°"  +  ^" ''"">'• 


2«  -+-  2    2  !  (2«  +  2)  (2«  +  4)    4  ! 


Q^-x      ^^  +  5  -^3    Y  (2^  +  7) (2^ +  9)  £l_ 

2«  +  2   3  !  (2«+  2)(2«  +  4)    5  ! 


♦  Finite  expressions  fory„  and  /_„  exist  when  «  is  of  the  form  /  4-  ^,  /  being 
an  integer.  These  are  multiples  of  tJj  and  tJj,  Art.  217,  respectively.  Substituting 
in  the  numerical  factors  the  values  of  the  corresponding  Gamma  functions,  which  are 

(2/>+i)! 

r(/+  !)  =  ip^\') r(/+  D  =  Jp+^p^  ^^> 

and,  taking  account  also,  in  the  case  of  A  +  ^,  of  the  factor  found  in  the  preceding 
foot-note,  we  find 

/.^,  _        C2/>)!        sin^[/>_^^^j^)]-cos.r[^_(^^^)] 
'^■'^"2/-i/!V^  ^>*+i 

and 


§  XVIL]     BESSELIAN  FUNCTION  OF  THE  SECOND  KIND.       239 


The  Besselian  Function  of  the  Second  Kind, 


221.  The  second  integral,  when  n  is  an  integer,  may  be 
found  by  the  process  employed  in  Arts.  175  and  176,  and  in 
similar  cases.     Thus,  changing  equation  (i),  Art.  215,  to 

{^  —  n){d--\-n  —  h)y  +  x^y  =  o, 

and  putting  ^  =  2^^^;r'«  +  *'',  the  relation  between  consecutive 
coefficients  is 


Ar=- 


(w  +  2r  —  «)  {m  -^  2r  ■\-  n  ■—  A') 


where  h^  is  put  for  h.     Making  m  =  n  and  m^—n^h  suc- 
cessively, we  have  the  integrals 


^„_^-.  . 

'     .0 

in-\-  2 

-K) 

2.4(2^  + 2- 

-/^' 

){z„  +  ^-h')    " 

and 

y-n^x- 

•n  +  A     J 

T^ 

+  h- 

X^ 

.h^)(^2n-2- 

■h) 

+  ... 

4_ 

x-'^ 

'  (^ 

H-/^- 

K). 

..(2« 

+  h-h^){2n 

—  2 

-h)...{. 

-h) 

.(, 

x^ 
{2n  -\-  2  -{•  h  — 

■h')(2+h) 

+  ...' 

Denoting  the  product  of  x*"  and  the  series  last  written  by  ^{h)^ 
we  have  j/'(o)  =  jk«,  and  the  complete  integral  y  =  A^yn  -f-  Boy^*,;, 
may  be  written 

y  =  Aoyn  +  ^0^+  f  (I  +  /^  log^  +  ...)[>'«  +  '^'A'(o)  +...], 


240  BESSEVS  EQUATION.  [Art.  221. 

where,  when  h  =  o, 


B^  - 


2.4  ..  .  2n{2n  —  2){27i  —  4)  ...  2  2^*^~^n\  {n  —  \)\ 


and  T  denotes  the  aggregate  of  terms  in  ^_„,  which  remain 
finite  when  ^  =  o.     We  have  therefore 

y^Ayn^-  BoT+  Byn  \ogx  +  B^\,\o)  +  .  .  . , 

and  may  take  as  the  second  integral,  when  ^  =  o, 

yn\o%x—  2««-^«!(«  —  i)!7'+«A'(o)- 

If  this  expression  be  divided  by  2«« !,  the  first  term  becomes 
y„log;ir;  denoting  the  quotient  by  F„,  and  developing  j/''(o)  as 
in  Art.  173,  we  have 

K=/JogA:-2«-»(«-i)!:r-«ri+-^^ 

L         «  — I  2« 

, I JC4_  I £«»2l__"l 

(«~i)(«  — 2)  24.2!      "*       (;?  — i)  !  2»'»-»(«  — i)!j 

2«+»«!Li.(«  + 1)\        n-\-i)2^ 

I.2.(«  + l)(«  +  2)  V  2        ;«+I        «4-2/24  J 

and  the  complete  integral  of  Bessel's  equation,  when  n  is  an 
integer  may  be  written 

y  =  AJn  +  BK^ 


§  >  VII.]  LEGENDRE'S  EQUATION.  24 1 

The  integral  F„  is  called  the  Besselian  function  of  the  second 
kind.-^ 

Legendre's  Equation. 
22J.   The  equation 

{1  —  x^)-^  —  2x^ +  n{n  +  i)y  =  o^   .     .     .     .  (i) 
ax^  ax 

or,  as  it  may  be  written, 

is  known  as  Legendres  EquatioUy  because,  when  n  is  an  integer, 
it  is  the  differential  equation  satisfied  by  the  nXh  member  of  a 
set  of  rational  integral  functions  of  x  known  as  the  Legendrean 
Coefficients.!  Particular  interest,  therefore,  attaches  to  the 
case  in  which  «  is  a  positive  integer;  and  it  is  to  be  noticed 


*  The  properties  of  the  Besselian  functions  are  discussed  in  Lommel's  "  Studien 
iiber  die  Bessel'schen  Functionen,"  Leipzig,  1868;  Todhunter's  "Treatise  on  La- 
place's Functions,  Lame's  Functions,  and  Bessel's  Functions,"  London,  1875,  etc. 

t  The  Legendrean  Coefficient  of  the  wth  order  is  the  coefficient  of  a«  in  the 
expansion  in  ascending  powers  of  a  of  the  expression 


y/{l  —  2ax  +  a2) 
and  is  denoted  by  Pn{x),  or  simply  by  P„.    It  is  readity  shown  that 

whence,  substituting  F=  ^.^anPn  and  equating  to  z.ero  the  coefficient  of  tt»,  we  find 
"When  X  =1,  V=  — i—  =  1  +  o  +  o*  +  .  .  . ;  hence  Pn{i)  =  i  for  all  values  of  n. 


242  LEGENDRES  EQUATION.  [Art.  222. 

that  this  includes  the  case  in  which  «  is  a  negative  integer;  for, 
if  in  that  case  we  put  —  n  =s  n'  +  i,  whence  —  («  -f  i)  =  «',  we 
shall  have  an  equation  of  the  same  form  in  which  n'  is  zero  or  a 
positive  integer. 

223.  When  written  in  the  t9-form,  Legendre's  equation  is 

t?(t^  -  i)^ -  a:='(t9  -  «)  (i>  +  «  4- 1)^  =  o, 

a  binomial  equation  in  which  both  terms  are  of  the  second 
degree  in  1%  Hence  the  equation  may  be  solved  in  series  pro- 
ceeding either  by  ascending  or  descending  powers  of  x.  Putting 
jy  =  S^-^r^*"^"**  we  have,  for  the  integrals  in  ascending  series, 


^'x  =  I  -  /«(«  -f  i)^+  «(«  -  2)(«  +  i){n  +  3)^-  . . ., 
2!  4! 


and 


^,=a:-(«-i)(;^  +  2)^'+(«~i)(«-3)(«  +  ^)(«+4)^-.... 
Again,  writing  the  equation  in  the  form 

and  putting  y  =  ^^ArX*"'^'',  we  have  the  integrals  in  descend- 
ing series 

\  2(2«—  l)  2.4(2«-- l)(2«  — 3)  / 

and 

y,  =  X-n-^  (l  +  (^+l)(^  +  2)^., 
\  2(2;?  +  3) 

^  (n  +  i)(n  -h  2)(n  -^  3)(n  -\-  4)^^_^  ^  ^  ^  \ 

2.4(2«  +  3)(2«-|-5)  '*'/ 


§  XVIL]  THE  LEGENDREAN  COEFFICIENTS.  243 


The  Legendrean  Coefficients, 

224.  When  «  is  a  positive  integer,  j^,  or  y^  is  a  finite  series 
according  as  n  is  even  or  odd  ;  and  in  either  case  J3  is  a  finite 
expression,  differing  ixomy^  ox  y^  only  by  a  constant  factor.  If 
y^  be  multiplied  by  the  constant 

{2n-  i){2n'-  3)  ...  I     Qj.        {2n)\ 
n\  2«(«!)"' 

the  resulting  integral  is  the  Legendrean  coefficient  of  the  «th 
order,  which  is  denoted  by  P„.  By  the  cancellation  of  common 
factors  in  the  numerators  and  denominators  of  the  coefficients, 
the  successive  values  of  Pn  may  be  written  as  follows  :  — 


2 


^  4.2  4.2  4.2' 


^  4.2  4.2  4.2     ' 

11^  6      2:1:5  4  J.  1:5:^  ,     5-3-i 
^6  -  6.4.2  ^     3 6.4.2^  "^  3  6.4.2^  ~  6.4.2' 


in  which  the  law  of  formation  of  the  coefficients  is  obvious.* 


*  The  constant  is  so  taken  that  the  definition  of  P„  given  above  agrees  with 
that  given  in  the  preceding  foot-note.  For,  putting  x  =  i,  and  forming  the  differ- 
ences of  the  successive  fractions  which  in  the  expressions  last  written  are  multipHed 
by  the  binomial  coefficients,  it  is  readily  shown  that  P„{i)=i,iot  all  values  of  n. 


244  I.EGENDRES  EQUATION.  [Art.  225. 

The  Second  Integral  when  n  is  an  Integer. 

225.   When  n  is  an  integer,  the  second  integral  of  Legendre's 
equation  admits  of  expression  in  a  finite  form. 
Assume 

y=zuP„-v, (i) 

where  u  and  v  are  functions  of  x.  By  substitution  in  equation 
(i),  Art.  222,  we  have 

«  j  (I  - x-)"^  -  2^^  j^n{n-^i)pA  +  2{i- x-)^  ^ 
K  ax  ax  )  ax  ax 

-\-Pn\  (1-^=')- 2x—  ^-(i-jc»)- — \-2x—-n{n-\-\)v^o, 

(.  dx"^  dx )  dx^  dx 

in  which  the  coefficient  of  u  vanishes,  because  P„  is  an  integral, 
and  that  of  P^  will  vanish  if  u  be  so  taken  that 

f  ^^d^u  du 

(l  —  XA- 2X—-  =  O. 

^  '  dx^  dx 

du 

This  condition  is  satisfied  if  we  take  (i  —  ;t:*)  — -  =  i,  whence 

dx 

"  =  *1°Sj3T' •    •    •  W 

the  equation  then  becomes 

and  we  shall  have  a  solution  of  Legendre's  equation  in  the 
assumed  form  (i),  if  v  is  determined  as  a  particular  integral  of 
this  equation. 


§  XVI I.]  THE   SECOND  INTEGRAL.  245 

Now,  since  /*„  is  a  rational  and  integral  algebraic  function 
of  the  «th  degree,  the  second  member  of  equation  (3)  is  an 
algebraic  function  of  the  («  —  i)th  degree ;  hence  the  particular 
integral  required  is  the  sum  of  those  of  several  equations  of  the 
form 

(i  —  x"^)-^  —  2x-^  +  n{n  +  i)y  =  axP^      .     .     .   (4) 
dx^  ax 

in  which  /  is  a  positive  integer  less  than  n.  Solving  equation 
(4)  in  descending  series,  the  particular  integral  is 

~       {J>-n){J>  +  n-\-i)\         {p-^n-i){p-n-2)^ 

+ /(/-i)(/-2)(/-3) ^-4  +  ..  V 

{p-\.n-i){p-\-n-z){p-n-2){p-n-^)  / 

which,  when  /  is  an  integer,  is  a  finite  series  containing  no 
negative  powers  of  x.  Thus  the  particular  integral  of  equation 
(4)  is  an  algebraic  function  of  x  of  the  pth.  degree,  and  that  of 
equation  (3)  is  an  algebraic  function  of  the  {n  —  i)th  degree. 
Denoting  this  function  by  i?„,  we  have  therefore  an  integral 
of  Legendre's  equation  of  the  form 


226.    Since 


(2«  =  iAiog^-ie« (5) 


X  —  1        X        3^3        gjcs 


the  product  ^Pn  log ,  when  developed  in  descending  series, 

X  """  1 

commences  with  the  term  containing  x*^~^',  and  as  Rn  contains 
no  terms  of  higher  degree,  the  development  of  Q„  cannot  con- 
tain x^.  It  follows  that,  putting  Q„  =  Ay^  -h  By^  where  y^  and 
y^  are  the  integrals  in  descending  series.  Art.  223,  we  must 


246  LEGENDRE'S  EQUATION.  [Art.  2261 

have  Qn  =  By^*  But  j4  commences  with  the  term x-'^-^y  we 
therefore  infer  that  in  the  product  above  mentioned  the  terms 
with  positive  exponents  are  the  same  as  those  of  /?„,  and  are 
cancelled  thereby  in  the  development  of  Qn,  while  the  terms 
with  negative  exponents  vanish  until  we  reach  the  term  Bx-*^^^. 
The  formation  of  the  required  terms  of  this  product  affords  a 
ready  method  of  calculating  R„.^( 

*  To  determine  the  value  of  B,  we  notice  that  equation  (3),  Art.  147,  gives,  for 
the  relation  between  the  integrals  Pn  and  Qn  of  Legendre's  equation, 

/.„^-^„^  =  _^l_, (r) 

dx  dx        I  ^  xi 

where  y4  is  a  definite  constant.     Substituting  from  equation  (5),  this  gives 

P„^  +  (;r»  -  1)  \Pn~^  ^Rn~^'\=  A. 
L       dx  dx  J 

Putting  x—i,  we  have  A=i,  because  P«(i)=  i,  and  Pn  and  Pn  being  rational 
integral  functions,  the  quantity  in  brackets  does  not  become  infinite.  'Now,  from  Art. 

224,  Pn  =    ^^^^'    J3 ;  substituting  this  value,  and  putting  A  =  i,  Qn  =  By^^,  equation 

(i)  becomes  B(^2n)  \l     dy^  ^      dy-A  _       i 

2«(« !  )2  V  ^  dx       ^^dx  )  ~  i^  x^' 

Developing  both  members  in  descending  powers,  and  comparing  the  first  terms,  we 
whence  z?_    2*'(n\y  . 


B: 


C2W+I) 


t  The  Legendrean  coefficients  are  sometimes  called  zonal  harmonics,  the  term 
spherical  harmonics  (in  French  and  German  treatises /onciions  spheriques  and  Kugel- 
functionen)  being  applied  to  a  more  general  class  of  functions  which  include  them. 
The  function  Qn  is  the  zonal  harmonic  of  the  second  kind.  Discussions  of  the 
properties  of  the  functions  Pn  and  Qn  will  be  found  in  Todhunter's  Treatise  "  On 
Laplace's  Functions,  Lame's  Functions,  and  Bessel's  Functions,"  London,  1875  ; 
Ferrers'  "Spherical  Harmonics,"  London,  1877;  Heine's  "Handbuch  der  Kugel- 
functionen,"  Berlin,  1878 ;  etc. 


§  XVII.]  EXAMPLES.  247 


Examples   XVII. 

Solve  the  following  differential  equations  :  — 

dy    ,  a^        .  (x  —  d)e^^~^-{- c(x  •{■  a^e-*^^'^ 

1.  -^4.j;2  =  — ,        s  y=- ^^ ~ •• 

dx  x^  x'^{e(^^~'^  4-  ce-<^^~'^) 

d^u  _  8 

2.  -; a^x    ^u  =  o, 

dx^ 

u  =  Axe-^''''~\i  +  3«;c-*)  +^^^3«^~^(i  -  s^^'^)- 

d'^u       2  du 

3.  ~ ~-^a^u  =  o,      u  =  Ae^^(i—ax) +Be-'*-^(i -i-ax). 

dx^      X  dx 

dx"^      X  dx  ^ 

d^u   ,   2  du  ,     ^  -,  cos  (x  —  a) 

d^u      4  du        2    — 
dx^      X  dx  ^ 

u  =:Ax-^e^'^(i  —  ax)  -\-  Bx-^g-^^(i  -{-  ax). 

d^u   ,  A  du  , 
*      dx^^  X  dx  ' 

u  =z  Ax-^{cosax  +  axsinax)  +Bx-^{sinax  —  axQOsax). 

^^2         -^       x^ 

y  =  Ax~^e"^{i  —  ax  -i-ia^x^)  +Bx-^e-^^{i  -\-  ax  +  ^a^x^). 

d^y  ,  6v 

y  =  C^-2[(3  —  «2^2)  COS  («;ic  +  a)  +  3;2;t:  sin  {fix  +  a)]. 


^48  RICCATrS  EQUATION,  ETC.  [Art.  226. 

v  =  C(;r   "COS h^    *sin 1. 

^  \  a  a    ] 

y  =  Cj>;~^[(i  —  ^^»)  cos  (^  +  a)  +  ^  sin  {x  +  a)]. 

13.  Show  that,  when  q  is  the  reciprocal  of  an  odd  integer,  the  integral 
of  Riccati's  equation, 

a^A:*?-*^  =  o, 

dx'' 

may  be  written  in  the  form 

14.  Show  that  for  all  values  of  n 

,        ,n  -\-  2  x^       (n  -Y  2){n  -\-  a;)  x-i 
■^      "^w  +  i  2!"^  (;z+i)(«  +  2)  3!"^  ••• 

^ajTa- Ljf 

1-^4.  ^  +  ^  ^  _  (^  +  2)(;?4-4)  ^   , 

■^«+  I    2!  («+  l)(«+2)    3!"^   *•• 

15.  Show  that  the  complete  integral  of  the  equation 

dx^        dx      x-^ 
may  be  written  in  the  form 

xy  =  A{2  —  qx)  ■\' Be-9^{2-\- qx). 


§  XVII.]  EXAMPLES.  249 

16.   If  in  Riccati's  equation  a^  =  —  i,  show  that  the  integral  may 
be  expressed  in  Besselian  functions. 


«=V.[^A(l.)  +  ^/_.(i^)] 


17.   Reduce  to  Bessel's  form  the  equation 

x-^'tl  4.  nx%  +  (^  +  cx'''^)y  -  o, 
dx^  ■         dx 

and  show  that  its  integral  in  Besselian  functions  is 


2m 


d^y 


20.  •^^+^+;'  =  0»  ;'  =  4/o(2^i)+^Ko(2^i). 

^2y   ,</)/,       ,  .cos ^2  sin;c2 

21.  ^-j-^  +  3^  +  4;<:3y  =  o,  y  =  A \- B -. 

dx^      ^  dx  ^  -^  x"^  x^ 

22.  Putting  u  =  <?«V(^^  +  A^)  ^  XPrh"-,  show  that 

^^        2    _  ^^  '^^^ 

and  thence  that  ijj+i  is  an  integral  of 

d^v        ,        p(p  +  i) 
dx^  x^ 


250  EXAMPLES.  [Art.  226. 

23.   Pm  and  Pn  being  Legendrean  coefficients,  show  that 

and  thence  that 

r  PnPmdx  =  o, 

except  when  m^n.    Also  show  that,  when  m'\-n  is  an  even  number, 
PxPrndx  =  o,  unless  m=i  n. 


§  XVIIL]  SIMULTANEOUS  EQUATIONS.  25 1 


CHAPTER   X. 

EQUATIONS   INVOLVING   MORE   THAN   TWO   VARIABLES. 

XVIIL 

Determinate  Systems  of  the  First  Order. 

227.  A  system  of  n  simultaneous  equations  between  n+  i 
variables  and  their  differentials  is  a  determinate  system  of  the 
first  order,  because  it  serves  to  determine  the  ratios  of  the 
n-\-  \  differentials  ;  so  that,  one  of  the  variables  being  taken 
as  independent,  the  others  vary  in  a  determinate  manner,  and 
may  therefore  be  regarded  as  functions  of  the  single  indepen- 
dent variable. 

A  determinate  system  involving  the  variables  Xy  y^  Zy  . . . 
may  be  written  in  the  symmetrical  form 

dx  _  dy  _  dz  _ 

X  "  Y~  Z  ' 

in  which  Xy  F,  Z,  . . .  may  be  any  functions  of  the  variables. 

228.  When  the  system  is  put  in  this  form,  we  may  consider 
the  several  equations  each  of  which  involves  two  of  the  differen- 
tials ;  if  one  of  these  contains  only  the  corresponding  variables, 
it  is  an  ordinary  differential  equation  between  two  variables,  and 
its  integration  gives  us  a  relation  between  these  two  variables. 
This  integral  may  be  used  to  eliminate  one  of  these  variables 
from  one  of  the  other  equations,  and  may  thus  enable  us  to 
obtain  another  equation   containing  only  two  variables ;    and 


252  SIMULTANEOUS  EQUATIONS.  [Art.  228. 

finally,  in  this  manner,  n  integral  equations  between  the  «  +  i 
variables.     Given,  for  example,  the  system 

dx  __dy  __dz  ,  v 

y  ~~  X  "  z* 

in  which  the  equation  involving  dx  and  dy  is  independent  of  z ; 

integrating  it,  we  have 

^2  —  ^2  =  « (2) 

Employing  this  to  eliminate  Xy  the  equation  involving  dy  and  ds 

becomes 

dy        _  dz 
sJiy^  +  a)"  z' 

and  the  integral  of  this  is 

y  +  sj{y^  +  a)  =  bz (3) 

The  integral  equations  (2)  and  (3)  containing  two  constants  of 
integration  constitute  the  complete  solution  of  the  given  system. 


Transformation  of  Variables. 

229.  A  system  of  differential  equations  given  in  the  sym- 
metrical form  is  readily  transformed  so  that  a  new  variable 
replaces  one  of  the  given  variables.  For  example,  when  there 
are  three  variables  x,  y,  and  z,  let  it  be  desired  to  replace  x  by 
a  new  variable  «,  a  given  function  of  Xy  j,  and  z.     We  have 

^  __  ^  _  ^  _  Xdx  +  t^-dy  +  vdz  ,  s 

where  X,  /x,  and  v  denote  any  arbitrary  multipliers.      Now,  ii 
being  a  given  function  of  Xy  j,  Zy 

J        du  1     .   du  J    .   du  J 

du  =  — dx  -\ dy  -\ dz. 

dx  dy  dz 


§  XVI 1 1.]  TRANSFORMATION  OF   VARIABLES.  253 

Hence,  if  A.,  /a,  v  be  taken  equal  to  the  partial  derivatives  of  2/, 
the  numerator  of  the  last  fraction  in  equation  (i)  is  du^  and 
denoting  the  denominator  by  U,  we  have 


dy  _dz  _du 


(2) 


in  which  F,  Z,  and  U  are  to  be  expressed  in  terms  of  j,  ^,  and 
ti  by  the  elimination  of  x. 

As  an  illustration,  in  the  example  of  the  preceding  article 

we  may  write 

dx  __  dy  ^dz  _dx  +  dy  ^ 
y^x~'z~'y-\-x' 

so  that,  taking  u=x  -^-y^    we  have  for  one  of  the  equations 

dz  _du 

z  '^  u* 

of  which  the  integral  is 

u  —  bz, 

which  is  equivalent  to  equation  (3)  of  the  preceding  article. 


Exact  Equations. 
230.   If  X,  \k,  V  in  equation  (i),  Art.  229,  be  so  taken  that 

\X+  fiY+vZ=o, 

we  shall  have 

kdx  H-  fidy  +  vdz  =  o. 

An  equation  derived  in  this  manner  may  be  exact,  and  thus  lead 
directly  to  an  integral  equation  containing  all  three  of  the 
variables. 


254  *  SIMULTANEOUS  EQUATIONS.  [Art.  23O. 

For  example,  if  the  given  equations  are 

^•^      =      dy     =  — ^— , (i) 

mz  —  ny      nx  —  Iz      ly  —  mx 

we  thus  obtain 

Idx  +  mdy  +  ndz  =  0, (2) 

and  also 

xdx  +  ydy  -\-  zdz  =  o (3) 

Each  of  these  is  an  exact  equation,  and  their  integration  gives 

ix  -\-  my  -\-  nz  =  a, (4) 

and 

^*+r +  2'  =  ^, (5) 

which  constitute  the  complete  solution  of  the  given  equations. 


The  Integrals  of  a  System, 

231.  Denoting  an  exact  equation  derived  as  in  the  preceding 
article  from  the  system 

dx  _^  _dz  XV 

X  ""  y"*z ^^^ 

by  du  —  o,  the  multipliers  A.,  /a,  v  are  the  partial  derivatives  of 
the  function  Uy  and  the  relation  connecting  them  is 

A-^  +  y^  +  z^  =  o (2) 

dx  dy  dz 

Hence,  if  a  function  u  satisfies  this  condition,  the  exact  equation 
du  =  o  is  derivable  from  the  system  (i),  and  its  integral 

u  —  a 

may  be  taken  as  one  of  the  two  equations  which  constitute  the 
solution. 


§  XVin.]  THE  INTEGRALS   OF  A   SYSTEM.  255 

An  equation  of  this  form  containing  but  one  constant  of 
integration  is  called  an  integral  of  the  system  in  contradistinc- 
tion from  an  integral  equation  which,  like  equation  (3),  Art.  228, 
contains  more  than  one  arbitrary  constant. 

Conversely,  if  ti  =  a  is  an  integral  of  the  system  (i),  the 
function  ti  must  satisfy  equation  (2) :  for  let  us  transform  the 
system  as  in  Art,  229 ;  then,  because  du  =^  o,  we  shall  have 
f/=  o,  which  is  equation  (2). 

232.  When  there  are  more  than  three  variables,  we  can 
derive  in  the  same  way  a  similar  condition  which  must  be  satis- 
fied by  the  partial  derivatives  of  the  function  «,  when  «  =  ^  is 
an  integral.  Thus  it  is  possible  to  verify  a  single  integral  of 
a  system  without  having  a  complete  solution.  The  complete 
solution  of  a  system  involving  n  -\-  \  variables  may  be  put  in 
the  form  of  a  system  of  n  integrals  corresponding  to  the  n 
arbitrary  constants.  The  number  of  integrals  is,  however,  in 
any  case  unlimited  ;  for  in  the  complete  solution  we  may  replace 
any  constant  by  any  function  of  the  several  constants.     Thus, 

let 

u  =1  a      ^    and  v  =  d 

be  two  independent  integrals  of  a  system  involving  three  varia- 
bles, and  let  <^  denote  any  function,  then 

<f>{u,v)  =  <i>{a,d)  =  C 

is  a  relation  between  x,  y,  z  and  the  arbitrary  constant  C,  and  is 
therefore  an  integral.  This  is,  in  fact,  the  general  expression 
for  the  integrals  of  the  system  of  which  u  —  a  and  v  =  b  are 
two  independent  integrals.  Accordingly,  it  will  be  found  that, 
if  u  and  v  are  functions  of  x  and  y  satisfying  equation  (2)  of  the 
preceding  article,  <^  (w,  v)  also  satisfies  that  equation,  <^  being 
an  arbitrary  function. 


256  SIMULTANEOUS  EQUATIONS  [Art.  233. 


Equations  of  Higher  Order  equivalent  to  Determinate  Systems 
of  the  First  Order, 

233.  An  equation  of  the  second  order  may  be  regarded  as 
equivalent  to  two  equations  of  the  first  order  between  Xj  y  and 
/,  one  of  which  is  that  which  defines  /,  namely, 

and  the  other  is  the  result  of  writing  ^  in  place  of  — ^  in  the 

dx  dx^ 

given  equation.     For  example,  the  system  equivalent  to  the 

equation 

which  is  solved  in  Art.  'j^y  is,  when  written  in  the  symmetrical 
form  of  Art.  227, 

'-l^dx^-% 

p  y 

in  which  the  equation  involving  ^  and  ^  is  independent  of  x, 
and  thus  directly  integrable. 

The  integrals  of  the  equivalent  system  are  the  same  as  the 
first  integrals  of  the  equation  of  the  second  order,  of  which  two, 
corresponding  to  the  constants  of  integration  employed,  may  be 
regarded  as  independent.  Compare  Art.  79.  The  complete 
integral  of  the  equation  of  the  second  order,  containing  as  *it 
does  both  constants  of  integration,  is  an  integral  equation,  but 
not  an  integral,  being  the  result  of  eliminating  the  variable  / 
either  before  or  after  a  second  integration.     Compare  Art.  82. 

In  like  manner,  an  equation  of  the  «th  order  is  equivalent  to 
a  system  of  n  equations  of  the  first  order,  between  n  -h  i  varia- 
bles.    Again,  two  simultaneous  equations  of  the  second  order 


§  XVIIL]  GEOMETRICAL   INTERPRETATION.  25/ 

between  three  variables  are  equivalent  to  a  system  of  four 
equations  of  the  first  order  between  five  variables,  and  so  on. 

Geometrical  Meaning  of  a  System  involving  Three  Variables. 

234.  Let  X,  y  and  z  be  regarded  as  the  rectangular  coor- 
dinates in  space  of  a  moving  point ;  then,  since  the  system  of 
differential  equations 

dx  _dy  _dz 
X  ~Y~Z 

determines  the  ratios  of  dx,  dy  and  dz,  it  determines  at  every 
instant  the  direction  in  which  the  point  (;r,  y,  z),  subject  to  the 
differential  equations,  is  moving.  Starting,  then,  from  any 
initial  point  A,  the  moving  point  will  describe  a  definite  line, 
and  any  two  equations  between  x,  y  and  z,  representing  two 
surfaces  of  which  this  line  is  the  intersection,  will  form  a  parti- 
cular solution.  If  we  take  a  point  not  on  the  line  thus  deter- 
mined for  a  new  initial  point,  we  shall  determine  another  line  in 
space  representing  another  particular  solution.  The  two  equa- 
tions forming  the  complete  solution  must  contain  two  arbitrary 
constants,  so  that  it  may  be  possible  to  give  any  initial  position 
to  {x,  y,  z).  The  entire  system  of  lines  representing  particular 
solutions  is  therefore  a  doubly  infinite  system  of  lines,  no  two 
of  which  can  intersect,  assuming  X,  Y  and  Z  to  be  one-valued 
functions,  because  at  each  position  there  is  but  one  direction  in 
which  the  point  {x,  y,  z)  can  move.  We  hence  infer  also  that 
the  constants  will  appear  only  in  the  first  degree. 

235.  Consider,  now,  the  complete  solution  as  given  by  two 
integral  equations  between  x,  y,  z  and  the  constants  a  and  d. 
The  surfaces  represented  determine  by  their  intersection  a  par- 
ticular line  of  the  system.  Let  the  constant  ^  pass  through  all 
possible  values,  while  a  remains  fixed  ;  then  at  least  one  of  the 
surfaces  moves,  and  the  intersection  describes  a  surface.     The 


2S8  SIMULTANEOUS  EQUATIONS.  [Art.  235. 

equation  of  this  surface  is  the  integral  corresponding  to  the  con- 
stant a ;  for  it  is  the  result  of  eliminating  b  from  the  two  equa- 
tions, and  is  thus  a  relation  between  x,  y,  z  and  a.  Hence,  an 
integral  represents  a  surface  passing  through  a  singly  infinite 
system  of  lines  selected  from  the  doubly, infinite  system,  and  of 
course  not  intersecting  any  of  the  other  lines  of  the  system.* 

If  a  and  b  both  vary  but  in  such  a  manner  that  C  =  <t>  (a,  b) 
remains  constant,  the  intersection  of  the  two  surfaces  describes 
the  surface  whose  equation  is  the  integral  corresponding  to  the 
constant  C.     Compare  Art.  232. 

236.  Thus,  in  the  example  given  in  Art.  230,  the  integral  (4) 
represents  a  plane  perpendicular  to  the  line 

I       m       n 

and  the  integral  (5)  represents  a  sphere  whose  centre  is  at  the 
origin.  The  intersection  of  the  plane  and  sphere  corresponding 
to  particular  values  of  the  constants  is  a  circle  having  its  centre 
upon,  and  its  plane  perpendicular  to,  the  fixed  line  (i). 

Hence  the  doubly  infinite  system  of  lines  represented  by  the 
differential  equations  (i).  Art.  230,  consists  of  the  circles  which 
have  this  line  for  axis ;  and  the  integrals  of  the  differential 
system  represent  all  surfaces  of  revolution  having  the  same  line 
for  axis. 

Examples  XVHI. 

Solve  the  following  systems  of  simultaneous  equations  :  — 

I.    ^=^==_^,  y^  +  ^^a,     \ogbx^i^n-^L 

X        z  y  z 


*  On  the  other  hand,  of  the  surface  represented  by  an  integral  equation,  we  can 
only  say  that  it  passes  through  a  particular  line  of  the  system. 


§  XVIIL] 


EXAMPLES. 


259 


'  dx        2X  _ 
dy  ,        ,    2X 


dx  dy 


X  —  -  -\ — 


X  -\-  y  =  be^. 


y  -\-  z      z-\-  X      X  -\-  y 

dx  _  dy  _   dz 

X^  —  y2  —  2*  2Xy         2XZ 

Idx        _        mdy      _ 


slix^y^z)^ 


—  y      X  —  z 


ndz 


y  =  az,     X*  -\-y^  +  z^  =  bz. 


i^x  H-  m^y  +  n^z  =  aj 


mn{y  —  z)       nl{z^x)       lm{x  —  yY      I'^x'' -^  ni^y^  ■\- n^z=.  b. 

r  adx       _        bdy       _        cdz  ax^  +  by^  4-  ^z'^  =  A, 

{b  —  c)yz       (c  —  a)zx       {a  —  b)xy       a'^x'^  +  b'^y^  -\-  c^^  =  B, 

^     dx  _dy  _  dz 

X        y       z  —  asjix"^ -\- y^  •\- z"^^ 

y  —  ax,     x^^'' =^  ^\z-\-sj{p(^ -\-y^-\-^)\ 
8.    Show  that  the  general  integral  of 

dx  _dy  _dz 
I        m       n 

represents  cylindrical  surfaces,  and  that  the  general  integral  of 

dx     _      dy     _     dz 
x-a~  y- P^  z^y 

represents  conical  surfaces. 


26o  SIMULTANEOUS  EQUATIONS  [Art.  237. 


XIX. 

Simultaneous  Linear  Equations. 

237.  We  have  seen  that  the  complete  solution  of  a  system 
of  simultaneous  equations  of  the  first  order  between  n  -\-  \ 
variables  consists  of  «  relations  between  the  n  -\-\  variables  and 
n  constants  of  integration.  Selecting  any  two  variables,  the 
elimination  of  the  remaining  n  —  \  variables  gives  a  rela- 
tion between  these  two  variables,  involving  in  general  the 
n  constants. 

We  may  also,  selecting  one  of  the  two  variables  as  inde- 
pendent, perform  the  elimination  before  the  integration,  the 
result  being  the  equation  of  the  «th  order,*  of  which  the  equa- 
tion just  mentioned  is  the  complete  integral. 

For  example,  in  the  case  of  three  variables,  x^  y  and  t,  if  we 
require  the  differential  equation  connecting  x  with  the  inde- 
pendent variable  /,  the  two  given  equations  are  to  be  regarded 

as   connecting  with   t  the   four  quantities  x,  y,  —  and  -^. 

Taking  their  derivatives  with  respect  to  t,  we  have  four  equa- 

dx    dv    d^x  d^v 

tions  containing  x,  r,  -r-,  -r-,    —r-  and    —r- ;   and  from  these 
^        -"    dt     dt     dt'  dt^  ' 

dv  dv^ 

four  we  can  eliminate  y,  -j-  and   -5-,  thus  obtaining  an  equa- 

at  dt^ 

tion  of  the  second  order,  in  which  x  is  the  dependent,  and  t  the 

independent  variable. 

238.  As  a  method  of  solution  the  process  is  particularly 
applicable  to  linear  equations  with  constant  coefficients,  since 

*  The  differential  equation  connecting  two  of  the  variables  may  be  of  a  lower 
order,  in  which  case  the  integral  relation  will  contain  fewer  than  n  constants.  For 
example,  one  of  the  equations  of  the  first  order  may  contain  only  two  variables,  as  in 
Art.  228,  and  then  the  integral  relation  will  contain  but  one  constant. 


§  XIX.]  LINEAR  SYSTEMS.  261 

in  that  case  we  have  a  direct  method  of  solving  the  resulting 
equations. 

For  example,  the  equations 


and 


f +  5^+;^  =  ^'    .......  (I) 


f^-x  +  sy  =  e^^ (2) 


are  linear  equations  with  constant  coefficients,  if  t  be  taken  as 
the  independent  variable.  Differentiating  the  first  equation, 
we  have 

and  since  — ^  does  not  occur  in  this  it  is  unnecessary  to  differ- 
entiate the  second.  Eliminating  -j-  and  j/  by  means  of  equa- 
tions  (2)  and  (i),  we  have 


The  complementary  function  is  (A  -\-  Bt)e~^*,  and  the  par- 
ticular integral  is  found  by  the  methods  of  section  X.  The 
resulting  value  of  x  is 

x={A  +  Bt)  e-^  +  ^et  -  ^e'^, 

and,  substituting  this  value  in  equation  (i),  we  find  without 
further  integration, 

y=-(A-hB+Bi)  e-^i+i^e-t  +  i^eK 


262  SIMULTANEOUS  EQUATIONS.  [Art.  239. 

239.  The  differentiation  and  elimination  required  in  the 
process  illustrated  above  are  more  expeditiously  performed  by 
the  symbolic  method.  For,  since  the  differentiation  is  indi- 
cated by  symbolic  multiplication  by  Z>,  the  equations  may  be 
treated  as  ordinary  algebraic  equations.  Moreover,  the  process 
is  the  same  if  one  or  both  the  equations  are  of  an  order  higher 
than  the  first. 

For  example,  the  system 

d'^y      dx  . 

dx    .       dy 

« 
when  written  symbolically,  is 

{2D^-  a,)y-Dx=2t, 
2Dy  -f  (4Z)  —  3)^  =  o. 

Eliminating  Xy  we  have,  in  the  determinant  notation, 

2D         4^  -  3  I  o     4^  —  3 

or 

(Z)-l)»(2Z>  +  3)^=2-f/, 

and  integrating, 

y=^{A+Bt)e^ +  Ce-i^-^t. 

The  value  of  x  is,  in  this  example,  most  readily  derived  from 

that  of  y  by  first  eliminating  Dx  from  the  given  equations,  thus 

obtaining 

{^D^  -h  2D-  i6)y  -  3^  =  8/, 

whence,  substituting  the  value  of  j/, 

X  =  <f'(6^  -  2A-  2Bt)  -  \Ce-¥-  \. 


§  XIX.]  NUMBER    OF  ARBITRARY  CONSTANTS.  263 

240.  Ordinarily,  in  finding  the  value  of  the  variable  first 
eliminated  it  is  necessary  to  perform  an  integration,  and,  when 
this  is  done,  the  new  constants  of  integration  are  not  arbitrary, 
but  must  be  determined  so  as  to  satisfy  the  given  equations. 
Thus,  if  in  the  preceding  example  the  value  of  x  had  been  de- 
rived from  the  first  of  the  given  equations,  after  substituting  the 
value  of  y,  it  would  have  contained  an  unknown  constant  in 
place  of  the  term  —  -J,  and  it  would  have  been  necessary  to 
substitute  in  the  second  equation  to  determine  the  value  of  this 
constant. 

The  value  of  x  may  also  be  derived  directly  from  the  result 
of  eliminating  y,  namely, 


2^)2  _  4         —D 

X  = 

2Z>^  — 4 

2/ 

2D         aD-z 

2D 

0 

The  complementary  functions  for  the  two  variables  will  then 
be  of  the  same  form,  and  will  involve  two  sets  of  constants. 
By  substituting  in  one  of  the  given  equations,  we  shall  have  an 
identity  in  which,  equating  to  zero  the  coefficients  of  the  several 
terms  of  the  complementary  function,  the  relations  between  the 
constants  may  be  determined. 

241.  The  number  of  constants  of  integration  which  enter 
the  solution  is  that  which  indicates  the  order  of  the  resultant 
equation.  This  number  is  not  necessarily  the  sum  of  the  in- 
dices of  the  orders  of  the  given  equations,  although  it  cannot 
exceed  this  sum ;  it  depends  upon  the  form  of  the  given  equa- 
tions, being,  as  the  process  shows,  the  index  of  the  degree  in  D 
of  the  determinant  of  the  first  members. 

Denoting  this  number  by  m,  the  values  of  the  n  dependent 
variables  contain  n  sets  of  m  constants,  of  which  one  set  is 
arbitrary.  Substituting  the  values  in  one  of  the  given  equa- 
tions, we  have  an  identity  giving  m  relations  between  the  con- 
stants ;  it  is  therefore  necessary  to  substitute  in  «  —  i  of  the 
given  equations  to  obtain  the  relations  between  the  constants. 


264 


SIMUL TANEOUS  EQUA TIONS. 


[Art.  242. 


Introduction  of  a  New   Variable. 

242.  The  solution  of  a  system  of  differential  equations  is 
sometimes  facilitated  by  the  introduction  of  a  new  variable,  in 
terms  of  which  we  then  seek  to  express  each  of  the  original 
variables.     Given,  for  example,  the  system 


dx      dy       dz  ,   >, 


where 


X-ax-^-by-^-cz-ird,         V=  a'x  +  b'y  +  c'z  +  d', 
Z=a"x-{-b"y-\-c"z-\-d"^ 

If  we  introduce  a  new  variable  t  by  assuming  dt  equal  to  the 
common  value  of  the  members  of  equation  (i),  we  shall  have 
the  system 

dx      dy       dz        ,.  .  . 

Z=y=z=*' ('> 

involving  four  variables,  which  is  linear  if  t  be  taken  as  the 
independent  variable.  Writing  the  equations  symbolically,  the 
system  is 


{a  —  D)x  -\- by  -\- ez  +  d   =0, 

a'x  +  {b'  -  D)y  +  c'z  +  ^'  =  o, 

a^^x  +  ^'>  +  (c"  -  D)z  -f-  ^"  =  o ; 


(3) 


whence 


a-D 


b'-D 


D 


d 

b 

c 

d} 

b 

— 

D 

c' 

d' 

/^" 

^"- 

D 

(4) 


§  XIX.] 


INTRODUCTION  OF  A  NEW  VARIABLE. 


265 


in  which  D  may  be  omitted  in  the  second  member  because  it 
contains  no  variable.     Denoting  the  roots  of  the  cubic 


a-D       b 
a'       b'-D 


(5) 


by  Xi,  Xa  and  A.3,  equation   (4)  and  the  similar  equations  for  y 
and  z  give 

y  =  A^e^-'  J^B'e^-'  +Ce^^'  +k'    L      .     .     .     .   (6) 
z  =  A"e^-'  +  ^"^^^'  +  C"/«'  H-  /^"  J 

in  which  ky  k\  k"  are  the  values  of  ;r,  /,  3  respectively,  which 
make  X  =  o,  V=o  and  Z  =  o. 

Substituting  these  values  in  the  first  of  equations  (3),  we 
have  one  of  the  three  equations  determining  k,  k'  and  k",  and 
for  the  constants  of  integration  the  three  relations, 

{a  -  K)A  +  bA'  +  ^A"  =  o, 
{a-\^)B-\-bB'-{-cB"=o, 
(a-X^)C-^bC'-{-cC"^o. 


In  like  manner,  substitution  in  each  of  the  other  equations 
gives  three  relations  between  the  constants,  making  in  all  nine 
relations,  of  which  six  are  independent.  The  three  relations 
between  A,  A^  and  ^"  are 

(^-Xx)^  +  ^^'  +  ^^"=o, 

a^A  +  (b'-K)A'-\-c'A"  =  o, 

a"A-\-b"A'-h{c"-X,)A"=o, 


266  SIMULTANEOUS  EQUATIONS.  [Art.  242. 

which  are  equivalent  to  two  equations  for  the  ratios  A  :A':A'\ 
since  their  determinant  vanishes  because  Aj  is  a  root  of  equa- 
tion (5). 

243.  The  introduction  of  a  new  variable,  as  in  the  preceding 
article,  introduces  a  new  constant  of  integration  into  the  system, 
but  this  constant  is  so  connected  with  the  new  variable  that  the 
relations  between  the  original  variables  obtained  by  eliminating 
the  new  variable  are  independent  also  of  this  constant.  Thus 
in  the  value  of  x,  equation  (6),  we  might  have  put  /  +  a  in  place 
of  /,  employing  only  two  other  constants ;  then  the  relations 
between  x,  y  and  2,  which  we  should  obtain  by  eliminating  /, 
would  obviously  contain  only  the  two  constants  last  mentioned. 


Examples   XIX. 
Solve  the  following  systems  of  linear  equations  :  — 

at  at 

dx  dy  ,. 

2.    =  — ^—  =  dt^ 

Zx—y      x-\-y 

x={A+  Bt)e^*,  y=(A-B-\-  Bt)e'^. 

3*    (Sy  +  9^)^*^  +  dy  +  dz  =  o,     (4JV  +  3^)^-^  -{- 2dy  —  dz  =  o, 

y  =  Ae-^+Be-7^^    z  =  —  ^Ae-"^  ^Be-?"^, 


4.  ^=^/=^, 

--  my  mx 


X  =  A  cos  mt  '\-B  sin  mt,     y  =  A  sin  mt  —  B  cos  mt. 


§  XIX.] 


EXAMPLES. 


267 


;.   a —- -\' n^y  =  e"^ ,     -f- 


S.   a-  +  n^y=^e',     -  +  «^  =  o, 


az=  —  nAe»^  +  nBe-""^  — 


«*—  I 


6.   ^  +  ^ay  =  o,      4^  -  m^x  =  o, 
dt^  ^  df^ 


mx  tux 


=  .^^(. 


y  =  g^  ( A^  sm A^cos 


y/2  y/2 


mx 


wa:        ^     .    mx 
cos ^,  sin  — - 


N/2 


7-    i 


dx   .      dy    .  .  . 

4  ^  +  9  ^  +  44^  +  49;^  =  ^> 

3^  +  7^  +  34^+38^'=^', 


X  =  ^^-"^  +  Be-^i  4-  -^//  -  ¥  -  -^^^ 
y=  —  Ae-*  +  4^^-  6'  —  ^i-t  +  V  +  ^<f'. 

^-  f +  f +  '^-^^  =  ^'     f  +  5^  +  3J'  =  o, 

y-Acost-\-Bs\nt,    x=  -  3il±:g cos /  +  ^  ~  3-^ gin /. 

^     d/'  dt  dt^  dt  ^        ' 


x  =  Aj.  cos  af  —  Ar,  sin  a/  +  B^  cos  /?/  —  B2  sin  yS/, 

^  =  ^2  cos  a.t  +  ^i  sin  a/  +  B^  cos)8/  +  ^1  sinyg/; 

where  a  and  /3  stand  for  --  n  ±  sj{n^  +  ^2). 


268 


SIMULTANEOUS  EQUATIONS.  [Art.  243. 


10. 


rdx, 
dt 

= 

«3^3 

«i^i» 

dx, 
dt 

= 

«i^i 

«2^3, 

dx^ 
dt 

= 

a^^ 

^3^3, 

Ax    4-   «2  ^2    4-   ^2 


a^a^A 


/^'  + 


a^a^B 


^        (Ax  +  ^3)  (^x  +  ^2)  (X2  +  ^3)  (^2  +  ^2)  ^3' 


where  X,  and  X2  are  the  roots  of 

A»  +  (^?i  +  «2  +  ^3)^.  H-  a^a2  +  «2«3  +  «3<?i  =  o. 


dx  dv 


a:  =  ^'(^  COS  ^  +  ^  sin  /), 
y  =  e^i[(A  -B)  cost  +  {A  +  B)  sin/]. 


12.    < 


dx   ,      dy    ,        ,  f 


—  2^  +  3j>^=  12  —  3<f^ 


X  =  y^<?-4^  COS  /  +  Be-'^i  sin/  +  fi ^'  -  ff , 
^=  -  (^+^)^-4^cos/+(^-^)<?-4'sin/-3^<f^  +  -j^. 


13.   /f+- 


-2^=/,       /^  +  ^  +  5;;=/», 


^=      At-^-\-   Bt-^-\'^tAr^t\ 
y^^At'^-^Bt-^-^t  +  ^t-. 


§  XIX.]  EXAMPLES.  269 


d^x       dy       o         o^      ^-^1      ^y 


y  =  (3^  -  2^  -  2^/).f2<  -  \Ce-'^i  -  ^. 

16.   Show  that  the  integrals  of  the  system 

^^ax^-by^c,        ±=.a^x  +  b^y  +  c\ 
at  dt 

are 

{a  +  m^a')  {x  -\- m^y)  -^  c -\-  m^c^  =  ^^^C^  +  '^i^')^^ 

{a  +  Wa^')  (^  +  m^y)  +  ^  +  w^^'  =  A^e^""  +  '''2«')^, 
where  ;«i  and  m^,  are  the  roots  of 

a^m"^  ■\-  {a  —  b^^tn  —  b  —  o\ 
and  obtain  a  similar  solution  for  the  system 

d'^^  I    z  ^^V  f      I    zr 

^/2  dfl 


270       EQUATIONS  INVOLVING  THREE  VARIABLES.      [Art.  244 

XX. 

Single  Differential  Equations  involving  more  than  Two  Variables. 

244.  When  the  number  of  differential  equations  connecting 
«  -f-  I  variables  is  less  than  «,  it  is  of  course  impossible  to  estab- 
lish n  integral  relations  between  the  variables.  We  shall  here 
consider  only  the  case  of  a  single  equation,  at  first  supposing 
the  number  of  variables  to  be  three;  and  we  shall  find  that  there 
does  not  always  exist  an  equivalent  single  integral  relation  be- 
tween the  variables. 

We  have  seen  that  when  there  are  two  differential  relations 
between  x,  y  and  z^  the  integrable  equations  which  separately 
furnish  the  two  independent  relations  between  the  variables 
are  generally  produced  by  the  combination  of  the  given  equa- 
tions. We  have  now  to  find  the  condition  under  which  a  single 
given  equation  is  thus  integrable,  and  the  meaning  of  an  equa- 
tion in  which  the  condition  is  not  fulfilled. 

The  Condition  of  Integrabiliiy. 

245.  The  given  equation  will  be  of  the  form 

Pdx -\- Qdy -\- Rdz  =^  o, (i) 

in  which  Py  Q  and  R  may  be  any  functions  of  Xy  y  and  z. 
If  there  be  an  integral  relation  between  Xy  y,  z  and  an  arbitrary 
constant  a  to  which  this  equation  is  equivalent,  let  it  be  put 
in  the  form 

so  that  a  shall  disappear  by  differentiation ;  then  the  differential 

equation  du  =  o,  or 

du  ,    I  du  ,    ,  du  , 

—  dx-\-  —  dy  +  —  dz  =  o, 

dx  dy  dz 


§  XX.]  THE   CONDITION  OF  INTEGRABIIITY.  2/1 

must  be  equivalent  to  equation  (i).  In  other  words,  if  the 
equation  is  integrable,  there  must  exist  a  function  of  x^  y  and  z 
whose  partial  derivatives  are  proportional  to  P,  Q  and  R  ; 
thus 

dx  dy  \      dz 

Now,  since  —-—-=---—,  etc.,  these  equations  give 
dy  dx      ax  dy 


\dy        dx  J  dx         dy 

JdQ_d^^^dj._Qdj.^ 
\dz        dy)  dy  dz 

/dR  _  dP\  ^ 
\dx        dz ) 


pit  _^^ 

dz  dx 


Multiplying  the  first  of  these  equations  by  i?,  the  second  by  P 
and  the  third  by  Qy  and  adding  the  results,  ju,  is  eliminated,  and 
we  have 

-(f-f)*<f-f)--(f-f)  "■■■<■) 

for  the  condition  under  which  the  equation  (i)  admits  of  an 
integral.* 

*  If  the  given  equation  is  exact,  the  three  equations  above  are  satisfied  by  ^=  i, 
and  each  of  the  binomials  in  equation  (2)  vanishes.  If  one  of  the  binomials  van- 
ishes while  equation  (2)  is  satisfied,  an  integrating  factor  which  is  a  function  of  one 
variable  only  exists,  and  in  this  case  /t  is  readily  determined. 

In  general,  if  /x  is  an  integrating  factor  and  «  =  a  is  the  corresponding  integral, 
F{u)  =  F(a),  where  F  is  any  function,  is  also  an  integral,  and  ixF'{u)  is  the  corre- 
sponding integrating  factor.  Thus  ijf{u)  is  the  general  expression  for  the  integrat- 
ing factor. 


272       EQUATIONS  INVOLVING  THREE   VARIABLES.       [Art.  246. 


Solution  of  the  Integrablc  Equation. 

246.    Supposing  the  condition  of  integrability  to  be  satisfied, 

if  in  the  integral  one  of  the  variables,  say  ^,  be  regarded  as 

constant,  the  corresponding  differential  equation  between  x  and  y 

will  be 

Pdx  -f  Qdy  —  o. 

Hence  the  complete  integral  of  this  equation  will  include  the 
integral  sought  if  the  constant  of  integration  be  regarded  as  a 
function  of  js.  Finally,  this  function  of  2  may  be  determined  by 
comparing  the  total  differential  equation  of  the  complete  inte- 
gral with  the  given  equation. 

Given,  for  example,  the  equation 

zydx  —  zxdy  —  y^dz  =  0, (i) 

in  which  P  =  zy,  Q  =  —  zx,  R  —  —y"",  and  the  condition  of  in- 
tegrability is  found  to  be  satisfied.     Treating  ^  as  a  constant, 

the  equation  becomes 

ydx  —  xdy  =0, (2) 

of  which  the  complete  integral  is 

x-Cy=o (3) 

This  isj  therefore,  the  integral  of  equation  (i),  C  being  inde- 
pendent of  X  and  y  but  involving  z.     Differentiating,  we  have 

dx  —  Cdy  —  ydC  =  o. 

Multiplying  by  zy  to  make  the  term  containing  dx  identical 
with  that  in  equation  (i),  the  coefficients  of  dy  are  identical  by 
virtue  of  equation  (3),  and  the  equations  agree  if  —  zy^dC='-y^dZf 
or 

Z 


§  XX.]         SOLUTION  OF  THE  INTEGRABLE  EQUATION.         273 

Hence  C  =  c  -^  log-s",  and  the  integral  (3)  becomes 

X  —  cy  —  y  log  0  =  0, 

which  is  the  integral  of  equation  (i). 

247.  It  is  to  be  noticed  that  the  possibility  of  obtaining  a 
differential  relation  between  C  and  ^,  independent  of  x  and  y, 
sufficiently  indicates  the  integrability  of  the  equation.  But  in 
some  cases  it  is  necessary,  in  order  to  obtain  such  an  equation, 
to  eliminate  the  other  variables  from  the  equation  containing 
dC  by  means  of  the  integral  itself.  Thus,  let  the  given  equa- 
tion be 

xdx  -f-  zdz  =  ^(7/2  —  x'^  —  z'^)dy (i) 

If  y  be  regarded  as  constant,  the  integral  is 

jc2  H-2==  C. (2) 

This  is  therefore  the  integral  of  equation  (i),  if  it  be  possible 
to  determine  C  as  3.  function  of  y.  Differentiating  and  com- 
paring with  the  given  equation,  we  find 

^dC=sJ{k^-x^-z^)dy, 

an  equation  containing  x  and  ^ ;  but,  eliminating  x  by  means  of 
equation  (2),  z  also  disappears,  and  we  have 

dC  , 

=  dy. 


2  s/{/i^  -  C) 
Hence 

sJ{h--C)  =  -y  +  c, 

and,  substituting  the  value  of  C  thus  determined  in  equation  (2), 
we  have,  for  the  integral  of  equation  (i), 


274       EQUATIONS  INVOLVING  THREE   VARIABLES.       [Art.  248. 


Separation  of  the  Variables, 

248.  When  it  is  possible  to  put  the  equation  in  such  a  form 
that  one  of  the  variables  occurs  only  in  an  exact  differential, 
the  equation  will,  if  intcgrable^  be  thus  rendered  exact.  Sup- 
pose, for  example,  that  it  can  be  written  in  the  form 

d7u -{- Sdx -\- Tdy  =i  o, (i) 

in  which  5  and  T  are  independent  of  z.  Now,  if  there  be  an 
integral,  it  may  be  put  in  the  form  z  =f{x,y) ;  hence,  also,  by 
substituting  this  value  of  z  in  the  expression  for  w  as  a  func- 
tion of  Xt  y  and  ^,  it  may  be  put  in  the  form 

w  -\-f^{x,y')  =  0 .     .     .  (2) 

The  differential  of  this  equation  must  be  identical  with  equa- 
tion (i),  because  the  terms  containing  dz  are  identical ;  there- 
fore, if  equation  (i)  be  integrable,  it  is  already  exact,  and  its 
integral  is  i 

w-\-  <i>{x,y)  =  c. 

In  fact,  the  condition  of  integrability,  Art.  245,  reduces  in 
this  case  to 

dS_dT^^ 
dy        dx 

which  is  the  same  as  the  condition  of  exactness  for  the  differ- 
ential expression  Sdx  -^^  Tdy.     See  Art.  25. 

249.  The  most  obvious  application  of  this  principle  is  to  the 
case  in  which  one  variable  can  be  entirely  separated  from  the 
other  two.  Thus  the  example  in  Art.  246  might  have  been 
solved  in  this  way ;  for,  dividing  by  zy^,  which  separates  the 
variable  Zy  it  becomes 

y  dx  —  xd'v       dz 
; =  0, 


§  XX.]  HOMOGENEOUS  EQUATIONS.  275 

an  exact  equation  of  which  the  integral  is 

X 

log  2  =  ^.  \ 

y 

Homogeneous  Equations, 

250.  In  the  case  of  a  homogeneous  equation  between  Xy 
y  and  z,  one  variable  can  be  separated  from  the  other  two  by 
means  of  a  transformation  of  the  same  form  as  that  employed 
in  the  corresponding  case  with  two  variables,  Art.  20.  For, 
putting 

X  ■=  zUy  y  —  zv, 

the  homogeneous  equation  may  be  written  in  the  form 

z**(f>(u,  v)dx  +  z"ij/(u,  v)dy  -{-  z"x{^,  v)dz  =  o ; 

and,  substituting 

£^x  =  zdu  -f-  udZy  dy  =  zdv  -f  vdz^ 

we  have 

z<^{u,  v)du  -f  z\\i  {u,  v)dv  +  [x(«)  ^)  +  U(^{u,  v)  +  v\\i{u,v)'\  dz  =  o. 

If  the  coefficient  of  dz  vanishes,  we  have  an  equation  between 
the  two  variables  u  and  v.     If  not,  the  equation  takes  the  form 

dz  <f>{u,  v)du  -f  \\i{u,  v)dv        _ 

z        x{u,v) -\-u<l>{u,v)  -{-z;if/(u,z>)~'    ' 

and,  in  accordance  with  Art.  248,  the  second  term  will  be  an 
exact  differential  if  the  given  equation  is  integrable. 

251.  As  an  example,  let  us  take  the  equation 

(jj;2  -\.yz-\-  z^)dx  +  (2=  +  zx  -{•  x^)dy  -\-  {x^  +  xy  +y')dz  =  o,  .  (i) 


2/6  EQUATIONS   CONTAINING  [Art.  25 1. 

which  will  be  found  to  satisfy  the  condition  of  integrability. 
Making  the  substitutions,  and  reducing,  we  have 

dz       (f"  -^  V  -\-  \)du  -\-  (^W^  -\-  u  -\-  \)dv  _ 
z  {u  -\-  If  +  \)  {uv  -\-  u  -\-  v)  ~~    ' 

Knowing  the  second  term  to  be  an  exact  differential,  we  in- 
tegrate it  at  once  with  respect  to  //,  and  obtain 

log  Z  -  log  ; — -  +  C  =  o, 

The  symmetry  of  this  equation  shows  that  6^  is  a  constant  and 
not  a  function  of  v :  thus  the  integral  of  equation  (i)  is 

xy  +yz  -\-  zx  =  {:{x  -{-y  -\-  z). 


Equations  containing  more  than  Three  Variables, 

252.    In  order  that  an  equation  of  the  form 

Pdx  4-  Qdy  +  Rdz  +  Tdt  =  o 

involving  four  variables  may  be  integrable,  it  must  obviously  be 
integrable  when  any  one  of  the  four  variables  is  made  constant. 
Thus,  regarding  z,  x  and  y  successively  as  constants,  equation 
(2),  Art.  245,  gives  the  three  conditions  of  integrability, 

-(f-f)--(f-f)-<f-f)=°' 
-(f-f)-<f-f)--(f-f)=°' 

\dx        dz)  \dt        dx)         \dz        dt ) 


§  XX.]  MORE   THAN  THREE    VARIABLES.  2// 

Again  regarding  /  as  constant,  we  have  the  condition 


\az        ay  J  \ax        az  J 


dP__dQ 

dy        dx 


but  this  is  not  an  independent  condition,  for  it  may  be  deduced 
by  multiplying  the  preceding  equations  by  R^  P  and  Q  respec- 
tively, and  adding  the  results. 

253*    In  general,  if  the  equation  contains  n  variables,  the 
number  of  conditions  of  the  above  form  which  we  can  write  is 

M    I'll    ___     t\    f^f,    II         2  I 

_i Li ^,  which  is  the  number  of  ways  we  can  select 

1.2.3 

three  out  of  the  n  variables.     But,  in  writing  the  independent 

conditions,  we  may  confine  our  attention  to  those  in  which  a 

selected  variable  occurs,  for  any  condition  not  containing  this 

variable  may  be  obtained  exactly  as  in  the  preceding  article 

from  three  of  those  which  do  contain  it.      Thus  the  number 

of  independent  conditions  is  - — — — '  ^    ~  "  ,  which  is  the  num- 

1.2 

ber  of  ways  we  can  select  two  out  of  the  n  —  i    remaining 

variables. 

254.    When  the  conditions  of  integrability  are  satisfied,  the 

integral  is  found,   as  in  the  case  of  three  variables,   by  first 

integrating  as  if  all  the  variables   except  two  were  constant, 

the  quantity  C  introduced  by  this  integration  being  a  function 

of  those  variables  which  were  taken  as  constants.     To  determine 

this  function  the  total  differential  of  the  result  is  compared  with 

the  given  equation.     The  result  either  determines  the  value  of 

dC  in  terms  of  these  last  variables  (in  which  case  dC  should  be 

an  exact  differential),  or  else  is  such  that  the  first  two  variables 

may  be  eliminated  simultaneously,  as  in  the  example  of  Art. 

247,  giving  an  integrable  equation  between  C  and  the  remaining 

variables. 


278       EQUATIONS  INVOLVING  THREE  VARIABLES.       [Art.  255. 

The  Non-Iniegrable  Equation, 

255.  In  an  equation  of  the  form 

Pdx  +  Qdy  +  Rdz  —  o 

the  variables  x^  y  and  z  may  have  any  simultaneous  values 
whatever ;  but,  for  each  set  of  values,  the  equation  imposes  a 
restriction  upon  the  relative  rates  of  variation  of  the  variables, 
that  is,  upon  the  ratios  of  dx,  dy  and  dz.  When  the  condition 
expressed  by  equation  (2),  Art.  245,  is  satisfied,  there  exists  an 
integral  equation  which,  for  each  of  the  sets  of  values  of  x,  y 
and  z  which  satisfy  it,  imposes  the  same  restriction  upon  their 
relative  rates  of  variation.  At  the  same  time  the  presence  of 
an  arbitrary  constant  makes  the  integral  sufficiently  general  to 
be  satisfied  by  any  simultaneous  values  of  x,  y  and  z. 

But,  when  the  condition  of  integrability  is  not  satisfied,  there 
is  no  such  integral  equation.  Two  integral  equations  will,  how- 
ever, constitute  a  particular  solution,  when,  for  each  set  of 
simultaneous  values  of  x,  y  and  z  which  satisfy  them,  the  ratios 
which  they  determine  for  dx,  dy  and  dz  satisfy,  in  connection 
with  these  values,  the  given  differential  equation. 

256.  If  one  of  the  two  integral  equations  is  assumed  in 
advance,  the  determination  of  the  particular  solutions  consistent 
with  the  assumed  equation  is  effected  by  solving  a  pair  of 
simultaneous  differential  equations,  namely,  the  given  equation 
and  the  result  of  differentiating  the  assumed  relation.  Geo- 
metrically the  problem  is  that  of  determining  the  lines  upon  a 
certain  surface  which  satisfy  the  given  differential  equation. 

For  example,  given  the  equation 

{\  -^  2a)xdx  ■\- yi^i  —  x)dy  ■\- zdz  ■=^  o     ....   (i) 
(which  it  will  be  found  does  not  satisfy  the  condition  of  inte- 


§  XX.]  THE  NON-INTEGRABLE  EQUATION.  2/9 

grability) ;  let  it  be  required  to  find  the  lines  on  the  surface  of 
the  sphere 

^2    ^j^;2    _|_22   =    ^2 ,  .      (2) 

such  that  a  point  moving  along  any  one  of  them  satisfies  equa- 
tion (i).     Differentiating  equation  (2),  we  have 

xdx -\- ydy -^  zdz  ^=i  o, (3) 

which  with  equation  (i)  forms  a  system  of  which  equation  (2) 
is  one  integral  and  a  second  integral  is  required.  Subtracting, 
we  have  an  equation  free  from  z,  namely, 

2axdx  —  xydy  =0, 

the  integral  of  which  is 

^»  =  4^^  +  C.      , (4) 

Hence  the  required  lines  are  those  whose  projections  upon  the 
plane  of  xy  are  the  parabolas  represented  by  equation  (4). 

257.  In  order  to  form  a  general  solution  of  a  non-integrable 
equation,  the  assumed  equation  must  contain  an  arbitrary  func- 
tion.    We  might,  for  example,  assume 

.y^/W, (I) 

where  /  is  arbitrary,  because  any  particular  solution  consisting 
of  two  relations  between  x,  y  and  z  might  be  put  in  the  form 
y  =if{x),  z  =  <l>{x).  If,  therefore,  we  determine  all  the  particu- 
lar solutions  consistent  with  equation  (i),  the  result  will,  when 
f  is  regarded  as  arbitrary,  include  all  the  particular  solutions. 
The  equation  which  completes  the  solution  will,  as  in  the  pre- 
ceding example,  be  found  by  integration,  and  will  therefore 
contain  an  arbitrary  constant  C,  to  which  a  special  value  must 


2 So       EQUATIONS  INVOLVING  THREE   VARIABLES,        [Art.  257. 

be  given  (as  well  as  a  special  form  to  the  function/)  in  order  to 
produce  a  given  particular  solution. 

258.   The  general  solution  of  the  equation 

Pdx  +  Qdy -{■  Rdz  z=  o (i) 

may  be  presented  in  quite  a  different  form,  which  is  due  to 
Monge,  depending  upon  a  special  mode  of  assuming  the  equa- 
tion containing  the  arbitrary  function. 

Let  /x  be  an  integrating  factor  of  the  equation 

Pdx  -h  Qdy  =  o 

when  z  is  regarded  as  a  constant,  and  let  V=  C  be  the  corre- 
sponding integral,  so  that 

dF=  fiFdx  -{-  fiQdy. 

Then,  in  the  first  place,  the  pair  of  equations 

z=^,  and  F=  C, (2) 

where  c  and  C  are  arbitrary  constants,  constitutes  a  class  of 
particular  solutions  of  (i).  Now,  for  the  general  solution,  let  us 
assume 

^=<^W (3) 

Differentiating,  we  have 

fxPdx-\-fiQdy-\-['^-<f>'{z)\dz  =  o,.     .     .     .   (4) 

which,  combined  with  equation  (i),  gives 

(^  -  <f>\^)  -  f^^y^ = o. (5) 


§  XX.]  MONGERS  SOLUTION.  28 1 

Hence,  if  F=  ^{z)  be  taken  as  one  of  the  relations  between  the 
variables,  we  must  have,  in  order  to  satisfy  equation  (i),  either 
dz  =  o,  or  else 

^^  -  <^\z)  -i^R^o •   .     .  (6) 

dz 

The  first  supposition  gives  z  —  c  and  V—<f>{c),  a  system  of 
solutions  of  the  form  (2) ;  the  second  constitutes,  in  connection 
with  equation  (3),  Moitges  solution. 

It  is  to  be  noticed  that  when  it  is  possible  to  determine  <^  so 
that  equation  (6)  is  identically  satisfied,  the  given  equation  is 
integrable,  and  V—<^{z)  is  its  integral.  But,  in  the  non-inte- 
grable  case,  <^  is  to  be  regarded  as  arbitrary. 

Monge's  solution  includes  all  solutions  excepting  those  of 
the  form  (2).  To  show  this,  it  is  only  necessary  to  notice  that, 
with  this  exception,  any  particular  solution  can  be  expressed  in 
the  form  xz=zf^(z),  y=f^{z)\  and,  substituting  these  values  in 
the  expression  for  V  as  a  function  of  x,  y  and  z,  we  have  an 
equation  of  the  form  V=<f>{z)  determining  the  form  of  <^  for 
the  particular  solution  in  question.  The  particular  solution  is 
therefore  among  those  determined  by  one  of  the  two  methods 
of  satisfying  equation  (5) ;  and,  as  it  is  not  of  the  form  (2),  it 
must  be  that  determined  by  equations  (3)  and  (6). 

The  distinction  between  this  solution  and  that  given  in  Art. 
257  is  further  explained  in  Art.  262  from  the  geometrical  point 
of  view. 

Geometrical  Meaning  of  a  Single  Differential  Equation  between 
Three  Variables. 

259.  Regarding  x,  y  and  z  as  the  rectangular  coordinates  of 
a  variable  point,  as  in  Art.  234,  the  single  equation 

Pdx  -f-  Qdy^  Rdz  =  0 (i) 


l32       EQUATIONS  INVOLVING  THREE   VARIABLES.       [Art.  259. 

expresses  that  the  point  {Xy  y^  d)  is  moving  in  some  direction,  of 
which  the  direction-cosines  /,  niy  n^  which  are  proportional  to 
dx%  dy  and  dzy  satisfy  the  condition 

Pl-\-Qm-\-Rn^Q (2) 

Consider  also  a  point  satisfying  the  simultaneous  equations 


dx  __dy  _dz 
P~  Q~  R' 


(3) 


and  therefore  moving  in  the  direction  whose  direction-cosines 
satisfy 

A  =  A  =  JL (4) 

P       Q       R  ^^^ 

Suppose  the  moving  points  which  satisfy  equations  (i)  and  (3) 
respectively  to  be  passing  through  the  same  fixed  point  A  ;  then 
P,  Q  and  R  have  the  same  values  for  each,  and  equations  (2) 
and  (4)  give 

l\  +  Mfi  -\-  nv  =  o, 

which  is  the  condition  expressing  that  the  directions  in  question 
are  at  right  angles.  We  have  seen,  in  Art.  234,  that  equations 
(3)  represent  a  system  of  lines,  there  being  one  line  of  the 
system  passing  through  any  given  point.  Hence  equation  (i) 
simply  restricts  a  point  to  move  in  such  a  manner  that  it  every- 
where cuts  orthogonally  the  system  of  lines  represented  by 
equations  (3),  which  we  may  call  the  auxiliary  system. 

260.  Now,  suppose  in  the  first  place  that  equation  (i)  is 
integrable.  The  integral  represents  a  system  of  surfaces  one 
of  which  passes  through  the  given  point  A.  This  surface  con- 
tains all  the  possible  paths  of  the  moving  point  which  pass 
through  A,  and  every  line  in  space  representing  a  particular 
solution  lies  in  some  one  of  the  surfaces  belonging  to  the  system. 


§  XX.]  GEOMETRICAL  INTERPRETATION.  283 

The  restriction  imposed  by  equation  (i)  is  in  this  case  completely 
expressed  by  a  single  equation. 

Every  member  of  the  system  of  surfaces  represented  by  the 
integral  cuts  the  auxiliary  system  of  lines  orthogonally,  so  that 
equation  (2),  Art.  245,  considered  with  reference  to  the  system 
of  lines  represented  by  equations  (3),  expresses  the  condition 
that  the  system  shall  admit  of  a  system  of  orthogonally  cutting 
surfaces. 

261.  On  the  other  hand,  when  the  condition  of  integrability 
is  not  satisfied,  the  possible  paths  of  the  moving  point  which 
pass  through  A  do  not  lie  in  any  one  surface,  the  auxiliary 
system  of  lines,  in  this  case,  not  admitting  of  orthogonally  cut- 
ting surfaces.* 

When,  as  in  the  example  of  Art.  256,  the  point  subject  to 
equation  (i)  is  in  addition  restricted  to  a  given  surface,  the 
auxiliary  lines  not  piercing  this  surface  orthogonally,  there  is  in 
general  at  each  point  but  one  direction  on  the  surface  in  which 


*  The  distinction  between  the  two  cases  may  be  further  elucidated  thus :  Select 
from  the  doubly  infinite  system  of  auxiliary  lines  those  which  pierce  a  given  plane  in 
any  closed  curve,  thus  forming  a  tubular  surface  of  which  the  lines  may  be  called  the 
elements.  Then,  in  the  first  case,  points  moving  on  the  tubular  surface  and  cutting 
the  elements  orthogonally  will  describe  closed  curves  ;  but,  in  the  second  case,  they 
will  describe  spirals. 

The  forces  of  a  conservative  system  afford  an  example  of  the  first  or  integrable 
case.  For,  if  X,  Y  and  Z  are  the  components,  in  the  directions  of  the  axes,  of  a 
force  whose  direction  and  magnitude  are  functions  of  x,  y  and  z,  the  lines  of  force 
are  those  whose  differential  equations  are 

dx  _  dy  _  dz 
X  ~  Y~  Z' 
The  equation 

Xdx  +  Ydy  +  Zdz  =  0 

will  be  satisfied  by  a  particle  moving  perpendicularly  to  the  lines  of  force,  so  that  no 
work  is  done  upon  it  by  the  force  ;  and  this  equation  is  integrable,  the  integral 
V=  C  being  the  equation  of  a  system  of  /eve^  surfaces  to  which  the  lines  of  force  are 
everywhere  normal. 


284      EQUATIONS  INVOLVING  THREE  VARIABLES.       [Art.  26 1. 

the  point  can  move  perpendicularly  to  the  auxiliary  lines.  We 
thus  have  a  singly  infinite  system  of  lines  on  the  given  surface, 
for  the  solution  of  the  restricted  problem. 

262.  In  a  general  solution  the  assumed  surface,  as,  for 
example,  the  cylindrical  surface  represented  by  equation  (i), 
Art.  257,  must  be  capable  of  passing  through  the  line  in  space 
representing  any  particular  solution  ;  and,  the  surface  being  thus 
properly  determined,  the  line  in  question  will  be  a  member  of 
the  singly  infinite  system  determined  upon  the  surface  by  the 
additional  integral  equation  found. 

The  peculiarity  of  the  general  solution  of  Art.  258  is  that 
the  assumed  surface  V—<^{z)  is  made  up  of  elements  which  are 
themselves  particular  solutions  of  a  certain  class.  We  still  have 
a  singly  infinite  system  of  particular  solutions  upon  the  assumed 
surface,  namely,  the  elements  just  mentioned.  But  upon  each 
surface  there  is  in  addition  the  unique  solution  determined  by 
equation  (6).  The  points  on  the  line  thus  determined  are  excep- 
tions to  the  general  rule,  mentioned  in  the  preceding  article, 
that  at  each  point  there  is  but  one  direction  on  the  surface  in 
which  a  point  can  move  perpendicularly  to  the  auxiliary  lines. 
The  line  is,  in  fact,  the  locus  of  the  points  at  which  the  auxiliary 
lines  pierce  the  surface  orthogonally. 

Examples  XX. 
Solve  the  following  integrable  equations  :  — 

1.  2{y-\-z)dx  ■\-{x-\-T^y-\-2z)dy-\-  {x-^y)dz  ==  o, 

(x+yy{y-\-z)  =  c. 

2.  (y  —  z)dx  +  2{x-\-  3y  —  z)dy-'  2{x-\-2y)dz  =  o, 

{x-^2y){y^zy  =  c. 

3.  {a  —  z)  {ydx  ■{- xdy) -\- xydz  =  o,  xy  =  ^{z  —  a). 


§  XX.]  EXAMPLES.  285 

4.  {y-\-afdx-\-zdy—{y-\-d)dz  —  o,  z  =  {x -\- c)  (^y  ■\- a) . 

5.  {ay  —  bz) dx  +  {cz  —  ax) dy  +  {bx  —  cy) dz  =  o, 

{ax  —  cz)  =  C(^_>'  —  (^2) . 

6.  ^jc  4-^H- (-^ +^  +  2+ 1)^0  =  o,  (jt:+_>'  +  2;)<f^  =  ^. 

7.  {y^  -\-yz)dx  +  (x0  4-  2^)^  -f-  {y^  —  ^7)^0  =  o, 

y{x-^z)  =  c{y-^z). 

8.  (::c2  +  z'^)  {xdx  +ydy  +  2^2)  +  (x""  -{-y^-^-  z'^y^{zdx  —  xdz)  —  o, 

(j^2_j_j^2_|_22)i_j_  tan-^- =  c. 

z 

9.  2{2y'^-\-yz  —  z;^)dx-\-x{/^y-\-z)dy-\-x{y—  2z)dz  —  o, 

x^{y-\-z){2y^z)z=c. 

10.  {x^y  —  j^'3  —  ^j'^^)^^  _^  (^j;2  _  :^3  —  ^jc:^^)^  -}-  {xy^^  +  ^^j)^^  =  o, 

X  +  z  .  y  +  z  _^ 
y  X 

11.  (2^2  _j_  2^j  -f-  2:rs2  +  i)dx  -\-  dy  +  2zdz  =  o, 

^^\y  +  z^  -^  x)  +  c  —  o. 

12.  (2^  4-^^  +  2xt  —  z)dx  -\-  2xydy  —  xdz  +  jjc^^/  =  o, 

^2  _j_  ^j^2  _|_  ^2/  —  ^0  =  ^. 

13-   ^(j^^  +  z)^^  +  ^(jV  +  2  +  i)^  +  tdz  —  {y  +  z)dt  —  o, 

{y  4-  z)e^+y  =  a. 

14.  2!(jv  +  z)^^  +  z{u  —  x)dy  +y(x  —  u)dz  +y{y  +  z)du  —  o, 

(;;  -\-z){u  -\-  c)  H-  z{x  —  u)  =  o. 

15.  Find  the  equation  which  expresses  the  solution  of 

dz  =  aydx  +  ddy 
when  we  assume  ^  =/{x). 


a{/{x)dx  +  ^/(;.)  4-  C. 


286       EQUATIONS  INVOLVING  THREE   VARIABLES.       [Art.  262. 

16.  Find  the  equation  which  determines  upon  the  ellipsoid 

55+y.  +  ^  =  ' 

the  lines  which  satisfy 

xdx  -\-ydy  ^cfi  -^^-^jdz  =  o. 

17.  Find  the  equations  which  determine  upon  the  sphere 

X^  +y^  4-  22  =  /^2 

the  lines  which  satisfy 

\x{x  —  a)  -hyiy  —  d)\dz=  {z  —  c) {xdx  +  ydy), 

z=  C,     and     ax  -\-  by  -\-  cz  =  k^, 

18.  Show  that,  for  the  differential  equation  of  Ex.  17,  the  auxiliary 
system  of  lines  consists  of  vertical  circles,  and  verify  geometrically  the 
results. 

19.  Give  the  general  solution  in  Monge's  form  of  the  equation 

zdx  +  xdy  -\-  ydz  =  o. 

y -\- z\ogx  ■=  <^{z),     x(^\z) -\- y^  xXogx. 

20.  Find  a  general  solution  of 

ydx  —  {x  —  z)  {dy  —  dz) . 

y-z^<^{x),    y=s(X''Z)<t>\x). 


§  XXL]  PARTIAL  DIFFERENTIAL  EQUATIONS.  287 


CHAPTER  XL 

PARTIAL  DIFFERENTIAL   EQUATIONS   OF   THE  FIRST   ORDER. 

XXI. 

Equations  involving  a  Single  Partial  Derivative, 

263.   An  equation  of  the  form 

Pdx  +  Qdy  +  Rdzz=Q (i) 

which  satisfies  the  condition  of  integrabihty  is  sometimes  called 
a  total  differential  equation,  because  it  gives  the  total  differ- 
ential of  one  of  the  variables  regarded  as  a  function  of  the 
other  two.  Thus,  if  x  and  y  be  the  independent  variables,  the 
equation  gives 

P  O 

dz  •=  —  —  dx  —-^ dy. 

or,  in  the  notation  of  partial  derivatives, 

dz  P  .  . 

Tx^-R' :  '^'^ 

and 

dy  R' ^^^ 

that  is  to  say,  we  have  each  of  the  partial  derivatives  of  2  given 
in  the  form  of  a  function  of  x,  y  and  z. 


288  PARTIAL  DIFFERENTIAL  EQUATIONS        [Art.  263. 

An  equation  of  the  form  (2)  or  (3),  giving  the  value  of  a 
single  partial  derivative,  or  more  generally  an  equation  giving 
a  relation  between  the  several  partial  derivatives  of  a  function 
of  two  or  more  independent  variables,  is  called  a  partial  differ- 
ential equation. 

264.  To  solve  a  partial  differential  equation  of  the  simple 
form  (2),  it  is  only  necessary  to  treat  it  as  an  ordinary  differential 
equation  between  x  and  z,  y  being  regarded  as  constant,  and 
an  unknown  function  of  y  taking  the  place  of  the  constant  of 
integration.  The  process  is  the  same  as  that  of  solving  the 
total  differential  equation,  see  Art.  246,  except  that  we  have  no 
means  of  determining  the  function  of  y^  which  accordingly 
remains  arbitrary.  Thus  the  general  solution  of  the  equation 
contains  an  arbitrary  function. 

Equations  of  the  First  Order  and  Degree. 

265.  Denoting  the  partial  derivatives  of  ^  by/  and  ^,  thus 

i)  -^  —  —  ^ 

dx^  dy ' 

a  partial  differential  equation  of  the  first  order,  in  which  z  is 
the  dependent  and  x  and  y  the  independent  variables,  is  a  rela- 
tion between/,  q,  x, y  and  z.  A  relation  between  x, y  and  ^  is 
a  particular  integral^  when  the  values  which  it  and  its  derived 
equations  determine  for  z,  p  and  q  in  terms  of  x  and  y  satisfy 
the  given  equation  identically.  We  shall  find  that,  as  in  the 
case  of  the  simple  class  of  equations  considered  in  the  preced- 
ing article,  the  most  general  solution  or  general  integral  con- 
tains an  arbitrary  function. 

266.  The  equation  of  the  first  order  and  degree  may  be 
written  in  the  form 

Fp  +  Qq  =  R, (i) 


§  XXL]  OF  THE  FIRST  ORDER  AND  DEGREE.  289 

where  P,  Q  and  R  are  functions  of  x,  y  and  z.  This  is  some- 
times called  the  Imear  equation,  the  term  linear,  in  this  case, 
referring  only  to/  and  q. 

L^'  u^a, (2) 

in  which  ti  is  a  function  of  x,  y  and  z,  and  «  is  a  constant,  be  an 
integral  of  equation  (i).  Taking  derivatives  with  respect  to 
X  and  y,  we  have 

du   .   du  .       ^  ^„  J         du   .   dti 

— +  — /  =  o,         and        —  +  —^  =  0; 

dx      dz  dy       dz 

and  substituting  the  values  of  /  and  q,  hence  derived  in  equa- 
tion (i),  we  obtain 

pf^  +  ef^+^f  =  0 (3) 

dx  dy  dz 

Therefore,  ii  7t  —  a  is  an  integral  of  equation  (i),  u\s  3.  function 
satisfying  equation  (3),*  and  conversely. 

But  we  have  seen  in  Art.  231  that  this  equation  is  satisfied 
by  the  function  ii  when  ?/  =  «  is  an  integral  of  the  system  of 
ordinary  differential  equations, 

dx      dy      dz  ,  ^ 

—  =  ^^  =  — (4) 

F       Q      R  ^^^ 

Hence  every  integral  of  the  system  (4)  is  also  an  integral  of 
equation  (i). 

Now,  it  was  shown  in  Art.  232,  that  if 

u  =  a  and  v  =  b 


*  It  follows  from  the  definition  of  an  integral  that  this  equation  is  either  an 
identity,  or  becomes  such  when  z  is  eliminated  from  it  by  means  of  equation  (2); 
but,  since  it  does  not  contain  the  constant  a  which  occurs  in  equation  (2),  the  former 
alternative  must  be  the  correct  one. 


290  PARTIAL  DIFFERENTIAL  EQUATIONS.        [Art.  266. 

are  two  independent  integrals  of  the  system  (4),  the  equation 

/(«,  v)  =  C 

includes  all  possible  integrals  of  the  system.  Hence  this  equa- 
tion, in  which  /  is  an  arbitrary  function,  is  the  general  integral 
of  equation  (i).  It  is  unnecessary  to  retain  an  arbitrary  con- 
stant since/ is  arbitrary;  in  fact,  solving  for  «,  the  equation  may 
be  written  in  the  form 

u  =  4>{v)y 

which  expresses  the  relation  between  x^  y  and  z  with  equal 
generality. 

Thus,  to  solve  the  linear  equation  (i),  we  find  two  inde- 
pendent integrals  of  the  system  (4)  in  the  forms  u  =  a,  v  =  b, 
and  then  put  u  =  <l>{v)y  where  <^  is  an  arbitrary  function.  This 
is  known  as  Lagrange  s  solution. 

267.  It  is  readily  seen  that  we  can  derive  in  like  manner 
the  general  integral  of  the  linear  partial  differential  equation 
containing  more  than  two  independent  variables.  Thus,  the 
equation  being 

...  (I) 


pf  +pj;  +. 

dx^           dx^ 

dx^ 

the  auxiliary  system  is 

dx^      dx^ 
P.~  P. 

dxn      dz 
~  Pu  ~R'   '    ' 

...  (2) 

and,  if  «i  =  Cr,  n^  =  c^,  .  .  .,  n„  =  c„  are  independent  integrals  of 
this  system,  the  general  integral  of  equation  (i)  may  be  written 

/{u^,  «2,  ...««)  =  o, (3) 

where  /  is  an  arbitrary  function.  If  an  insufficient  number  of 
integrals  of  the  system  (2)  is  known,  any  one  of  them,  or  an 
equation  involving  an  arbitrary  function  of  two  or  more  of  the 
quantities  u^,  u^,  .  .  .,  u„  constitutes  a  particular  mtegral  of 
equation  (i). 


§  XXL]  THE  LAGRANGEAN  LINES.  29 1 


Geometrical  Illustration  of  Lagrange'' s  Solution. 

268.  The  system  of  ordinary  differential  equations  emp/oyed 
in  Lagrange's  process  are  sometimes  called  Lagrange  s  equations. 
In  the  case  of  two  independent  variables  they  represent  a  doubly 
infinite  system  of  lines,  which  may  be  called  the  Lagrangean 
lines.  We  have  seen  in  Art.  235  that  every  integral  of  the 
differential  system  represents  a  surface  passing  through  Hues 
of  the  system,  and  not  intersecting  any  of  them.  It  follows, 
therefore,  that  the  partial  differential  equation 

is  satisfied  by  the  equation  of  every  surface  that  passes  through 
lines  of  the  system  represented  by  Lagrange's  equations 

dx  _dy  _dz  ^ 

and  the  general  integral  is  the  general  equation  of  the  surfaces 
passing  through  lines  of  the  system. 
Given,  for  example,  the  equation 

{mz  —  ny)p -\- {nx  —  lz)q  —  1}  ^  mx,     .     .     .     •  (i) 

for  which  Lagrange's  equations  are 


dx       __       dy      _       dz 
mz  —  ny      nx  —  Iz      ly  —  mx 


(2) 


The  integrals  of  this  system  were  found,  in  Art.  230,  to  be 

Ix  +  my  -\-  nz  =  a, 

and 

x^ -\- y^ -\- z^  =  3 ; 

and,  as  stated  in  Art.  236,  the  lines  represented  being  circles 
having  a  fixed  line  as  axis,  every  integral  of  the  system  (2) 


292  PARTIAL  DIFFERENTIAL  EQUATIONS.        [Art.  268. 

-    , , 

represents  a  surface  of  revolution  having  the  same  line  as  axis. 
Thus  the  general  integral  of  equation  (i),  which  is 

/X  + my-hnz  =  <f){x^ -^-y^ +  z^), (3) 

represents  all  the  surfaces  of  revolution  of  which  the  line 

X  _  y  _  z 
I  ~~  m      n 
is  the  axis. 

269.    It  was  shown  in  Art.  260  that,  when 

Pdx  ^  Qdy -^  Rdz  =  o (i) 

is  the  differential  equation  of  a  system  of  surfaces,  the  system 
of  lines  represented  by 

dx^iil^dz  .. 

P       Q      R ^  ^ 

cuts  these  surfaces  orthogonally.  It  follows  that  the  surfaces 
represented  by  the  general  integral  of 

Fp+Qg  =  R, 

which  pass  through  the  lines  of  the  system  (2),  cut  the  surfaces 
of  the  system  (i)  orthogonally.  Hence,  as  first  shown  by 
Lagrange,*  if  the  equation  of  a  system  of  surfaces  containing 
one  parameter  c  be  put  in  the  form 

the  surfaces  which  cut  the  system  orthogonally  are  all  included 
in 

*  CEuvres  de  Lagrange,  vol.  iv.  p.  628;  vol.  v.  p.  560. 


§  XXL]  COMPLETE  AND   GENERAL  PRIMLTLVES.  293 

where  ii  —  a  and  v  =  b  are  two  independent  integrals  of 


dx 

dv 

dz 

. 

dV 

dV 

dl^ 

dx 

dy 

dz 

The  Complete  and  General  Primitives. 

270.  If,  in  an  equation  containing  x,  y  and  z,  z  be  regarded 
as  a  function  of  x  and  7,  we  may,  by  differentiation  with  respect 
to  X  and/,  obtain  equations  involving/  and  q  respectively ;  and 
by  the  combination  of  the  given  and  the  two  derived  equations 
we  can  derive  a  variety  of  partial  differential  equations  satisfied 
by  the  given  equation.  If  the  given  equation  contains  two 
arbitrary  constants,  their  elimination  leads  to  a  definite  differ- 
ential equation  of  the  first  order  independent  of  these  constants, 
and  of  this  equation  the  given  equation  is  called  a  complete 
primitive. 

Given,  for  example,  the  equation 

s=  a{x-\-y)  -\-b (i) 

By  differentiation  we  have  p  =  a,  and  q  =  a,  hence 

/  =  ^ (2) 

is  the  only  equation  of  the  first  order  independent  of  a  and  by 
which  can  be  derived  from  equation  (i).  Hence  equation  (i) 
is  a  complete  primitive  of  equation  (2).  We  do  not  say  the 
complete  primitive,  because  the  general  solution  ofp  =  q  is 

z=./(x-\-y), (3) 

and  therefore  any  equation  of  this  form  containing  two  arbitrary 
constants  is  a  complete  primitive  ofp  =  q.  In  fact,  equation  (3) 
gives/  =/'{x  +  j),  q  =/'(x  -{-j/),  whence  p  =  q.     The  equation 


594 


PARTIAL  DIFFERENTIAL  EQUATIONS.        [Art.  270. 


from  which  a  given  partial  differential  equatioti  can  be  obtained 
by  the  elimination  of  an  arbitrary  function  is  called  its  general 
primitive;  thus  equation  (3)  is  the  general  primitive  of  /  =  ^. 

271.   The  most  general  equation  between  :r,  j  and  z,  contain- 
ing one  arbitrary  function,  may  be  written  in  the  form 

f{u,  v)=o, (i) 

where  u  and  v  are  given  functions  of  x,  y  and  z.  Regarding  s 
as  a  function  of  x  and  j,  the  derived  equations  are 

!^r^ + !^  J + ^r-^ + ^^1 = o, 

du\_dx      dz    J       dv  \_dx      dz    J 

^f!^  +  ^^^l  +  f^f;^  +  !*  J  =  o. 

du  \_dy       dz    J      dv  \_dy       dz     J 


and 


The  result  of  eliminating  the  ratio  -i-  :  -J-  may  be  written  in 

du     dv 

the  form 


du         du 
dx      ^~dz 

dv    ,    ^dv 

r  P  — 

dx      ^  dz 


du   ,  du 

dy   .  dz 

dv_.  dv^ 

dy  dz 


o. 


Of  the  four  determinants  formed  by  the  partial  columns,  that 
containing/^  as  a  factor  vanishes,  and  we  have 


du 

du 

du 

du 

du 

du 

dx 
dv 

dy 
dv 

+  P 

dz 
dv 

dy 
dv 

^q 

dx 

dv 

dz 
dv 

dx 

dy 

dz 

dy 

dx 

dz 

an  equation  of  the  form 


Pp+Qq^R, 


§  XXI.] 


THE   GENERAL  PRIMITIVE. 


295 


in  which 


P  = 


du 

du 

du 

du 

du 

du 

dv 

dz 

dz 

dx 

dx 

dy 

->      <2  = 

,      R^ 

dv 

dv 

dv 

dv 

dv 

dv 

dy 

dz 

dz 

dx 

dx 

dy 

It  thus  appears  that  the  equation  of  which  the  general  primitive 
contains  a  single  arbitrary  function  is  linear  with  respect  to 
p  and  q. 

7ri2.  The  values  of  P,  Q  and  R  above  are  called  the  Jacob- 
ians  of  u  and  v  with  respect  to  y  and  ^,  z  and  Xy  x  and  y 
respectively,  and  are  denoted  thus, 


P^ 


(i{u,  v) 
d{y,zy 


d{u,  v) 
d{z,  x)' 


R 


d(u,  v) 
d(x,y) 


The  Jacobian  vanishes  when  ?/  and  v  are  not  independent  func- 
tions of  the  variables  expressed  in  the  denominator,  thus  R 
vanishes  if  either  ?/  or  ^  is  a  function  of  z  only.  Again,  P,  Q 
and  R  all  vanish  if  21  is  expressible  as  a  function  of  v.  In  this 
last  case  equation  (i)  is,  in  fact,  reducible  to  v  =  c,  which  con- 
tains no  arbitrary  function. 

When  P,  Q  and  R  are  given,  the  functions  7C  and  v  must  be 
such  that  their  Jacobians  are  proportional  to  P,  Q  and  R, 
Now,  if  we  put 


and  V  =■  b, 


we  shall  have 


du  ■,     .  du  ,    ,  du  J 

—  dx  -\ dy  -\ dz 

dx  dy  dz 


dv   J     ,   dv   y    .   dv   7 

—  dx  ■\ dy  -\ dz  =  o\ 

dx  dy  dz 


296  PARTIAL  DIFFERENTIAL  EQUATIONS.        [Art.  272. 

whence,  solving  for  the  ratios  dx-.dy.dzy^^  have 

dx      __      dy      _      dz 
d{u,  v)       d{u,  v)      d{u,  v) 
d{y,z)       d{z,x)       d{x,y) 

Hence  we  shall  have  found  proper  values  of  //  and  v  ii  21  —  a 
and  z;  =  ^  are  integrals  of 

dx  _dy  _dz 
'p'"Q~  R  ' 

We  have  thus  another  proof  of  Lagrange's  solution  of  the  linear 
equation. 

273.  In  like  manner,  if  there  be  n  independent  variables 
Xrj  x^j  .  .  .,  x„y  and  one  dependent  variable  2,  we  can  eliminate 
the  arbitrary  function /from  the  equation 

f{u^,  u^  .  .  .Un)  =  o, 

in  which  «x>  ^2?  •  •  •>  ^^»  are  n  independent  given  functions  of  the 
variables.  In  the  result  of  elimination  the  coefficient  of  the 
products  of  any  two  or  more  of  the  partial  derivatives  will 
vanish,  and  we  shall  have  an  equation  linear  in  these  deriva- 
tives, that  is  an  equation  of  the  form 

/'i/i  +  /'^A  +  •  •  •  +  Pnpn  =  R. 

Moreover,  each  of  the  coefficients  P^,  P^,  .  .  .,  P„  and  R  will 
be  the  Jacobians  of  u^y  ti^,  .  .  .,  ?/„  with  respect  to  71  of  the 
variables,  and  the  simultaneous  ordinary  equations  derived  from 

?^j  =  C-iy  U2  ^  ^aj 


.,?/«  =  c„  will  be 

dx-i       dx2                  dXn 

P^           P.                             Pn 

dz 
-R' 

where  /*i,  Pa,  ,  .  .,  P„  and  R  are  the  same  Jacobians. 


§  XXL]  EXAMPLES.  297 


Examples  XXI. 

Solve  the  following  partial  differential  equations  :  — 

dz 

1.  y- 2x  —  2z—y=Oy  X  •\-y  -\-  z^y'^^ipc), 

2.  psj{y^-x^)=y,  z=y^m-^--^^{y). 

3.  lpA-mq==i,  z=j  +  <f>{/y  —  mx). 
A'p-\-q  —  nz,  z  =  e"^<l)(x—y). 

5.  xp  +  yq  =  nz,  '  zz=x"(f>l^ 

6.  yp -\- xq  =  z,  z=  {x -{-y)(}i(x^ —y''). 

7.  {y^x  —  2x'^)p -}- (2y^  —  x^y)  q  =  gz{x^ —y^), 

^"y    \y^    -^v 

8.  xzp  +  yzq  =  xy,  z^  =  xy  +  ffi\  ^ 


9.  x^p-xyq-{-y^  =  o,  z  =  =^  +  cf>{xy). 

10.  zp-\-yq  =  x,  X  +  z  =ycf>(x^  —  z^). 

11.  xp  +  zq'-\-y  =  o, 

12.  {y  +  z)p+{z-^x)q  =  x+y, 


y 

II.    xp  +  zq  -\-y  =  o,  tan-^^  =  log:i:  + (^(jV^ +02). 


-A.^-y 


{z-y)sl{x-^y^-z):==<^ 

y  —  X       X         \   xy  j 
14.    x{y-z)p-^y{z-x)q=zz{x^y),  xyz  =  cf>{x +y -\-z). 

15-   ^-^=7:^'  (^+j)log2  =  :i:  +  <^(^-f>'). 


298  PARTIAL  DIFFERENTIAL  EQUATIONS.        [Art.  273. 

16.   z  —  xp  —  yq^  a\j{x'^  ■\- y^ -\- z'^), 

37.    {y-^x)p-\-{y-x)q  =  z, 

z  =  v/(^»  4-r)<^[tan-^^  +  ilog (^»  +r)]- 

18.  J2/  +  xyq  —  nxz,  z  —  y"cfi {x^  —  y''). 

19.  xy^p  —  y^^  +  ^•^■3^  =  o»  log^  = h  ^{p^y)' 

20.  (6"  —  :vOA  +  ("S*  -  ^a)/^  +  ...  +'(5  -  ^„)/«  =  6*  —  0, 
where  6"  =  ^i  +  ^2  +  ...  +  ^«  +  2, 

<^l'S«(^x  -  ^),  ^«(^»  -  ^),  .  .  .,  S^{xn  -  2)  J  =  o. 
dz    ,      dz    .    .dz  ,    xy 

dx        dy         dt  t  .    ^xy         ^,(y     i\ 

22.  Find  a  common  integral  of  the  equations 

py  =  qx     and    /^  +  ^j  =  2. 

z  =  csl{x^  ^-J^'^)• 

23.  Show  that  ^3  +  j;3  4-  03  —  3Jlr^'s  =  /^  is  a  surface  of  revolution, 
and  find  its  axis. 

24.  If  «  =  o  and  V  =  o  are  particular  integrals  of  a  linear  partial 
differential  equation,  show  that  every  other  integral  <^  =  o  satisfies  the 
equation 

d{<i>,  u,  v)  ^^ 
d{x,  y,  z)         ' 

25.  Determine  the  surfaces  which  cut  orthogonally  the  system  of 
similar  ellipsoids  ,         ,. 

m^      n^  \  z         z  J 

26.  Determine  the  surfaces  of  the  second  order  which  cut  orthogo- 
nally  the  spheres  :,^  +  y' +  z^  ^  2ax. 


x'^  -\-y^  +  z^  =  2by  -f-  2cz. 


§  XXIL]         EQUATIONS  NOT   OF  THE  FIRST  DEGREE.         299 

XXII. 

The  Non-Linear  Equation  of  the  First  Order, 

274.  We  have  seen  in  Art.  270  that  a  partial  differential 
equation  of  the  first  order  may  be  derived  from  a  given  primi- 
tive by  the  elimination  of  two  arbitrary  constants.  Such  a 
primitive  constitutes  a  complete  integral  of  the  differential 
equation  ;  but,  when  the  resulting  equation  is  linear,  the  general 
solution  contains  an  arbitrary  function  which  imparts  a  gen- 
erality infinitely  transcending  that  produced  by  the  presence  of 
arbitrary  constants  or  parameters.  The  surfaces  represented 
by  a  complete  integral  constitute  a  doubly  infinite  system  of 
surfaces  of  the  same  kind,  while  the  more  general  class  of  sur- 
faces represented  by  the  general  integral  is  said  to  form  a 
family  of  surfaces.  Thus,  in  the  example  given  in  Art.  270, 
the  complete  integral  (i)  represents  the  doubly  infinite  system 
of  planes  parallel  to  a  fixed  line ;  and  the  general  integral  (3) 
represents  the  family  of  cyHndrical  surfaces  whose  elements  are 
parallel  to  the  same  fixed  line. 

275.  The  differential  equation  derived  from  a  complete  prim- 
itive may  be  non-linear.     For  example,  if,  in  the  primitive, 

{x-hy -\-(^y-kY  +  z-^c% (i) 

h  and  k  are  regarded  as  arbitrary  parameters,  the  resulting 
differential  equation  is 

2^/'  +  ^'  +  0  =^S (2) 

which  is  not  linear  with  respect  to  /  and  q.  Equation  (i)  is 
therefore  a  complete  integral  of  equation  (2).  Geometrically  it 
represents  a  doubly  infinite  system  of  equal  spheres  having 
their  centres  in  the  plane  of  xy.     It  will  be  shown,  however,  in 


300        EQUATIONS  NOT   OF  THE  FIRST  DEGREE.         [Art.  275. 

the  following  articles,  that  the  geometrical  representation  of  the 
general  integral  of  a  non-linear  equation  is  a  family  of  surfaces 
equally  general  with  that  representing  the  general  integral  of  a 
linear  equation.  But,  since  it  has  been  shown  in  Art.  271  that  a 
primitive  containing  an  arbitrary  function  gives  rise  in  all  cases 
to  a  linear  equation,  it  is  obvious  that  the  general  integral  of  a 
non-linear  differential. equation  cannot  be  expressed  by  a  single 
equation.* 

The  System  of  Characteristics. 

276.    A  partial  differential  equation  of  the  first  order,  con- 
taining two  independent  variables,  is  of  the  form 

F{x,y,z,p,  q)  =  0 (i) 

Let 

z  =  <l>{x,y), (2) 

whence 

^  =  S'  ^=t^ <3) 

be  an  integral ;  then  these  values  of  ^,  /  and  g  satisfy  equation 
(i)  identically.  If  -r,  j  and  £•  be  regarded  as  the  coordinates  of 
a  point,  equation  (2)  represents  a  surface.  A  set  of  correspond- 
ing values  of  x,  j/,  z,  p  and  q  determine  not  only  a  point  upon 
the  surface,  but  the  direction  of  the  tangent  plane  at  that  point, 
and  are  said  to  determine  an  element  of  the  surface.  If  we  per 
mit  X  and  y  to  vary  simultaneously  in  any  manner,  the  corre- 
sponding consecutive  elements  of  surface  determine  a  linear 


*  The  surfaces  of  the  same  family  are  generated  by  the  motion  of  a  curve  in 
.space,  when  arbitrary  relati'jns  exist  between  its  parameters.  The  simplest  case  is 
that  in  which  there  are  but  two  parameters  ;  the  two  equations  of  the  curve  can  then 
be  put  in  the  form  u  =  d,  "—  C2;  and,  if  /(f  1,  ^2)  =  o  is  the  relation  between  the 
parameters,  /(u,  t^)  =  o  is  the  general  equation  of  the  family.  This  case,  therefore, 
corresponds  to  the  linear  differential  equation.  See  Salmon's  "  Geometry  of  Three 
Dimensions,"  Dublin,  1874,  PP  37^  e^  se^. 


§  XXII.]  THE   SYSTEM  OF  CHARACTERISTICS.  30I 

elemeftt  of  surface ;  that  is,  a  line  upon  the  surface  together 
with  the  direction  of  the  tangent  plane  at  each  point  of  the 
line. 

The  linear  element  thus  determined  upon  the  surface  (2)  will 
in  general  depend  upon  the  form  of  the  function  ^  ;  but  it  will 
now  be  shown  that,  starting  from  any  initial  point  upon  the 
surface,  there  exists  one  linear  element  which  is  independent  of 
the  form  of  <^,  provided  only  that  equation  (i)  is  satisfied,  so 
that  every  integral  surface  which  passes  through  the  initial  ele- 
ment must  contain  the  entire  linear  element. 

277.    Let  the  partial  derivatives  of  /^be  denoted  as  follows  : 

—  =  x     —  =  y      —  =  z      —  =  F      —z=o 

dx  dy  ^         dz  '         dp  '         dq 

Since  z,  p  and  q  are  functions  of  x  and  y,  the  derivatives  of 
equation  (i)  with  respect  to  x  andjj/  give 

X  +  Z/-fPg  +  (2j  =  o, (4) 

Y^Zq^P^±  +  Q^  =0 (5) 

dy  dy 

Now  let  X  and  y  vary  simultaneously  in  such  a  way  that 

then,  because  fot  every  point  moving  in  the  surface 

dz  =  pdx  +  qdy, 


we  have  also 

dz 


^^    pP^qQ (7) 


302  EQUATIONS  NOT  OF   THE  FIRST  DEGREE.      [Art.  277. 

Equations  (6)  and  (7)  give 

dx  _dy^  _        dz 

The  values  of  /  and  q  in  these  equations  being  given  in  terms 
of  X  and  J,  by  equations  (3),  they  form  a  differential  system  for 
the  variables  x,  y  and  z.  Starting  from  any  initial  point  {xo,yo,  ^o), 
this  system  determines  a  line  in  space  ;  and,  supposing  the 
initial  point  to  be  taken  on  the  surface  (2),  this  line  lies  upon 
that  surface. 

Now,  substituting  from  equation  (6),  and  remembering  that 

dq  __   d^'z  _  dp 
dx      dxdy      dy^ 

equation  (4)  becomes 

dx  dt       dy  dt 
whence 

g=-X-Z/ (8) 


In  like  manner,  equation  (5)  gives 

f^  =  -Y-Zi (9) 

Equations  (6),  (7),  (8)  and  (9)  now  give 

dx  ^dy  ^        dz         ^  dp       ^  dq  .. 

P        Q       pP+qQ  X  +  Zp  Y-^Zq'      '      ^     ^ 

a  complete  differential  system  for  the  five  variables  x,  y^ 
Zy  p  and  q.  Starting  from  any  initial  element  of  surface 
(^o>^o> 'S'oj/w  ^o),   this   system   determines   a  linear  element  of 


§  XXIL]  THE  SYSTEM  OF  CHARACTERISTICS.  303 

surface,  and  supposing  the  initial  element  to  be  taken  on  the 
surface  (2),  the  entire  linear  element  lies  upon  that  surface. 

Now  the  system  (10)  is  independent  of  the  form  of  the  func- 
tion </>,  and  the  only  restriction  upon  the  initial  element  is  that 
it  must  satisfy  equation  (i)  ;  it  follows  that  every  integral  sur- 
face which  contains  the  initial  element  contains  the  entire  linear 
element.  This  linear  element,  depending  only  upon  the  form  of 
equation  (i),  is  called  a  characteristic  of  the  partial  differential 
equation.  Through  every  element  which  satisfies  equation  (i) 
there  passes  a  characteristic* 

278.  A  complete  solution  of  the  system  (10)  consists  of  four 
integrals  in  the  form  of  relations  between  x,  y,  z,  p  and  q.  Mul- 
tiplying the  terms  of  the  several  fractions  by  X,  Y,  Z,  —P  and 
—  Q,  respectively,  we  obtain  the  exact  equation  dF=o,  of  which 
F=  C  is  the  integral.  But  it  is  obvious  that,  in  order  to  confine 
our  attention  to  the  characteristics  of  the  given  equation,  we 
must  take  C=o.  Thus  the  original  equation  is  to  be  taken  as 
one  of  the  integrals  of  the  characteristic  system.  The  other 
three  integrals  introduce  three  arbitrary  constants.  Hence  the 
characteristics  form  a  triply  infinite  system. 

For  example,  in  the  case  of  the  equation  given  in  Art.  275, 
which  may  be  written 

F  =  p--\-q--'-+i=o, (i) 


2  C"^ 

X  =  o,   Y—Oy  Z— — ,  P=2pj  Q  =  2qy  and  the  equations  of 
the  characteristic  are 


*  In  like  manner,  when  there  are  n  independent  variables,  a  set  of  values  of 
•^i,  X3,  .  .  .,  xn,  z,  Px,  P2,  .  .  .,  /«,  v^^hich  satisfies  the  differential  equation,  is  called  an 
element  of  its  integral,  and  the  consecutive  series  of  elements  determined  as  above 
are  said  to  form  a  characteristic.  See  Jordan's  "Cours  d'Analyse,"  Paris,  1887,  vol 
iii.,  pp.  318  ^^  seq. 


304         EQUATIONS  NOT  OF  THE  FIRST  DEGREE.       [Art.  2/8. 

^  —  ^  —       ^g       __  z^dp  __  z^dq  ,   . 

p^  q  ~  p^  +  q^~        c^p  ~        c^q ^^^ 

Of  this  system,  equation  (i)  is  an  integral  ;  the  relation  be- 
tween dp  and  dq  gives  a  second  integral  which  may  be  written 
in  the  form 

^=/tana (3) 

The  values  of/  and  q  derived  from  equations  (i)  and  (3)  are 

p  =  co.J^^L^^, (4) 

Z 

^=sinaV^ ^, (5) 

z 

and  these  equations  may  be  taken  as  two  of  the  integrals,  in 
place  of  equations  (i)  and  (3).  Substituting  these  values  in  the 
relations  between  dx  and  dy,  dx  and  dz  respectively,  we  obtain, 
for  the  other  two  integrals, 

y  —  X  tana  +  a;, (6) 

and 

{x%tza-\- by  =  c^ —  z"^ (7) 

These  last  equations  determine,  for  given  values  of  a,  a  and 
b,  the  characteristic  considered  merely  as  a  line,  and  then  equa- 
tions (4)  and  (5)  determine  at  each  point  the  direction  of  the 
element,  that  is  to  say,  the  direction  of  a  plane  tangent  to  every 
integral  surface  which  passes  through  the  characteristic. 


The  General  Integral. 

279.  It  follows  from  Art.  2^^  that  every  integral  surface 
contains  a  singly  infinite  system  of  characteristics,  so  that  if 
we  make  the   initial  element  of  a  characteristic  describe  an 


§  XXI I.]  THE   GENERAL  INTEGRAL.  305 

arbitrary  line  upon  the  surface  (the  Hnear  element  of  surface 
along  the  line  determining  at  each  point  the  values  of /o  and  q^, 
the  locus  of  the  variable  characteristic  will  be  the  integral  sur- 
face. Moreover,  if  we  take  an  arbitrary  line  in  space  for  the 
path  of  the  initial  point,  it  is  possible  so  to  determine  p^  and  q^ 
at  each  point  that  the  characteristic  shall  generate  an  integral 
surface.     For  this  purpose,  we  must  have  in  the  first  place, 

^{xo,yo,Zo,po,qo)  =  o (i) 

Again,  since  the  path  of  the  initial  point  is  to  lie  in  the  surface, 
so  that 

dzo  =  podxo  +  qodyo, 

taking  the  differential  equations  of  the  arbitrary  curve  to  be 

dxo  _  dyo  _  dZo  .^ 

~L~M~'N' ^'^ 

we  must  have 

N  =  PoL-\-qoM, (3) 

where  Z,  M  and  N  are  functions  of  Xo,  jTo  and  Zq.  Geometrically, 
this  last  equation  expresses  the  condition  that  the  initial  ele- 
ment must  be  so  taken  that  the  plane  tangent  to  the  surface 
shall  contain  the  line  tangent  to  the  arbitrary  curve. 

The  general  integral  may  now  be  defined  as  representing 
the  family  of  surfaces  generated  by  a  variable  characteristic 
having  its  motion  thus  directed  by  an  arbitrary  curve.* 


*  That  the  surface  thus  generated  is  necessarily  an  integral  will  be  seen  in  the 
following  articles  to  result  from  the  existence  of  a  complete  integral.  The  analytical 
proof  requires  that  it  be  shown  that,  for  a  point  moving  in  the  surface,  we  have  always 

dz  =  pdx  +  qdy, 

where  p  and  q  are  given  by  the  equations  of  the  characteristic.     If  the  common 
value  of  each  member  of  the  equations  (2)  be  denoted  by  dr,  the  variation  of  t  moves 


306        EQUATIONS  NOT  OF  THE  FIRST  DEGREE.        [Art.  279. 

In  the  case  of  the  linear  equation,  when  the  characteristics 
become  the  Lagrangean  lines,  the  values  of  po  and  q^  are  still 
those  which  satisfy  equations  (i)  and  (3)  ;  but  they  need  not  be 
considered,  because  there  is  but  one  Lagrangean  line  through 
each  point. 

Derivation  of  a  Complete  Integral  from  the  Equations  of  the 
Characteristic. 

280.  The  four  integrals  of  the  characteristic  system  contain 
X,  y,  z,  /,  qy  and  three  constants.  We  may  therefore  obtain,  by 
elimination  if  necessary,  a  relation  between  x,y,  2  and  two  of  the 
constants.  Every  such  equation  represents,  for  any  fixed  values 
of  the  constants,  a  surface  passing  through  a  singly  infinite  sys- 
tem of  characteristics,  but  not  in  general  a  system  of  the  kind 
considered  in  Art.  279,  so  that  the  equation  is  not  in  general  an 
integral  of   the   partial   differential  equation.      It  will  now  be 

the  characteristic,  and  that  of  t  [^dt  being,  as  iti  Art.  277,  the  common  value  of  each 
member  of  equations  (10)]  moves  a  point  along  the  characteristic.  The  motion  of  a 
point  along  the  surface  then  depends  upon  the  two  independent  variables  /  and  t. 
Then,  since 

dz='^-^dt^^-^dr,         dx=^-^dt^^dr,  dy=^dt-\-^dr, 

dt  dr  dt  dr  -^      di  dr 

and  the  equations  of  the  characteristic  give 


it  remains  only  to  prove  that 


or  that 


dt     ^  dt      ^  dt 


dz       .dx  .      dy 


dz       .dx         dy       tt     ^ 
dr         dr         dr 


Letting  /=  o  correspond  to  the  initial  point,  the  condition  dzo  =  podxo  +  qodyo  shows 
that  the  corresponding  value  of  U  is  zero,  tjvat  is  Uq  =  o.     Consider  now  the  value 

of  — .    This  IS 

dt  ^=^_^  ^_^^_^^_    ^ 

dt       dtdr      dt  dr         dtdr      dt  dr         dtdr 


§  XXIL]      DETERMINATION  OF  A  COMPLETE  INTEGRAL.     307 

shown  how  we  may  find  such  an  integral,  that  is  to  say,  since 
two  arbitrary  constants  occur,  a  complete  integral  of  the  given 
equation. 

Suppose  one  integral  of  the  characteristic  system,  in  addition 
to  the  original  equation  F=  o,  to  have  been  found.  Let  a  denote 
the  constant  of  integration  introduced,  and  consider  the  values 
of  p  and  q  in  terms  of  Xy  y,  z  and  a  determined  by  these  equa- 
tions. Now,  in  a  complete  solution  of  the  characteristic  system, 
each  characteristic  is  particularized  by  a  special  value  for  each 
of  the  three  constants  of  integration.  We  may  distinguish  those 
in  which  a  has  the  special  value  a^,  as  the  aj-characteristics  ; 
these  constitute  a  doubly  infinite  system  of  linear  elements  of 
surface,  which  together  include  all  the  point  elements  deter- 
mined by  the  above-mentioned  values  of  /  and  q,  when  the  par- 
ticular value  ttj  is  assigned  to  a. 

Now  these  aj-characteristics  lie  upon  a  system  of  integral 
surfaces.     To  show  this,  consider  a  transverse  plane  of  refer- 


But 

dH  ^  d  dz^dp  dx  d^x       dqdy  dy  . 

dtdr      dr  dt      dr  dt      ^  drdt      dr  dt       ^  drdt ' 

hence  » 

dU  _dp  dx  .dg  dy  _dp  dx  _d^  d^^ 

dt  ~  dr  dt       dr  dt       dt  dr      dt  dr 

Substituting  from  the  equations  of  the  characteristic,  this  becomes 

dt  dr  dr  dr  dr  dr  dr 

or,  since  Zdz  ■\-  Xdx  +  Ydy  -J-  Pdp  -f-  Qdq  =  o, 

^=-Z'^-\-pZ^  +  qZ^  =  ^ZCI. 
dt  dr  dr  dr 

The  integration  of  this  gives 

U^  Ce-J^Zdt^ 

and,  putting  ^=0,  we  have  C=  Uq=o;  hence,  so  long  as  the  exponential  remains 
finite,  U=  o,  which  was  to  be  proved.  See  Jordan's  "  Course  d' Analyse,"  vol.  iii., 
p.  Z^l' 


308        EQUATIONS  NOT   OF   THE  FIRST  DEGREE.        [Art.  280. 

ence.  This  is  pierced  at  each  point  by  one  of  the  ai-characteris- 
tics,  and  at  the  point  the  element,  which  we  may  take  as  the 
initial  element  of  the  characteristic,  determines  in  the  plane  of 
reference  a  direction.  If,  starting  from  any  position  in  the 
plane  of  reference,  the  initial  point  moves  in  the  direction  thus 
defined,  it  describes  a  determinate  curve  in  that  plane,  and 
the  corresponding  characteristic  generates  an  integral  surface. 
Varying  the  initial  position  in  the  plane  of  reference,  we  have  a 
singly  infinite  system  of  curves  in  that  plane,  and  a  singly  infi- 
nite system  of  integral  surfaces. 

We  have  thus  a  system  of  surfaces  at  every  point  of  which 
the  values  of/  and  q  are  the  values  above  mentioned  which 
involve  a,.  Hence,  if  these  values  be  substituted  in  the  equa- 
tion 

dz  =-pdx  -j-  qdy 

(which,  it  will  be  noticed,  is,  by  Art.  277,  one  of  the  differential 
equations  of  the  characteristic  system),  we  shall  have  an  equa- 
tion true  at  every  point  of  this  system  of  surfaces ;  in  other 
words,  we  shall  have  the  differential  equation  of  the  system.* 

The  integral  of  this  equation  will  contain  a  second  constant 
of  integration /8 ;  when  both  constants  are  regarded  as  arbitrary, 
it  represents  a  doubly  infinite  system  of  surfaces  containing  the 
entire  system  of  characteristics,  and  is  a  complete  integral. 

281.  As  an  illustration,  let  us  resume  the  example  of  Art. 
278.  Substitution  of  the  values  of  /  and  q,  equations  (4)  and 
(5),  in  dz  —pdx  +  qdy,  gives 

zdz  J  ,     7     • 

=  dx  cos  a  -\-  dy  sin  a. 


sJ{c^-2^) 


*  It  follows  that  the  equation  thus  found  is  always  integrable.  This  would,  of 
course,  not  be  generally  true  if  the  values  of  /  and  (/  simply  satisfied  the  equation 
F=o.  The  early  researches  in  partial  differential  equations  were  directed  to  the 
discovery  of  values  of  /  and  ^  which  satisfied  F=  o  and  at  the  same  time  rendered 
dz  =  pdx  +  ^dy  integrable.     See  Art.  294. 


§  XXIL]      DETERMINATION  OF  A  COMPLETE  INTEGRAL.     309 

whence,  integrating,  we  have 

2=  +  (jt:  cos  a  +  7  sin  a  H-  /8)2  =  c"^, 
which  is  therefore  a  complete  integral  of  the  given  equation 

2^/"  +  ^"  +  i)  =  ^^ 

This  complete  integral  represents  a  right  circular  cylinder  of 
radius  ^,  having  its  axis  in  the  plane  of  xy ;  and  since  equation 
(6),  Art.  278,  represents  a  plane  perpendicular  to  the  axis,  we 
see  that  the  characteristics  in  this  example  are  equal  vertical 
circles,  with  their  centres  in  the  plane  of  xy,  regarded  as  elements 
of  right  cylinders. 

It  follows  that  the  general  integral  represents  the  family  of 
surfaces  generated  by  a  circle  of  radius  c,  moving  with  its  centre 
in,  and  its  plane  normal  to,  an  arbitrary  curve  in  the  plane  of 
xy.  The  surfaces  included  in  the  complete  integral  just  found  are 
those  described  when  the  arbitrary  path  of  the  centre  is  taken 
'^  a  straigfht  line. 


Relation  of  the  General  to  the  Complete  Integral. 

282.  Since  all  the  integral  surfaces  which  pass  through  a 
given  characteristic  touch  one  another  along  the  characteristic, 
and  the  surfaces  included  in  a  complete  integral  contain  all  the 
characteristics,  it  follows  that  every  integral  surface  touches  at 
each  of  its  points  the  surface  corresponding  to  a  particular  pair 
of  values  of  a  and  y8  in  the  equation  of  the  complete  integral. 
The  series  of  surfaces  which  touch  a  given  integral  surface  cor- 
responds to  a  definite  relation  between  ^  and  a,  say  jS  =  <^  (a)  ; 
thus  the  given  integral  is  the  envelope  of  the  system  of  surfaces 
selected  from  the  complete  integral  by  putting  ^=<f>{a)  and  so 
obtaining  an  equation  containing  a  single  arbitrary  parameter. 


3IO       EQUATIONS  NOT  OF  THE  FIRST  DEGREE.        [Art.  282 

The  equation  of  the  envelope  of  a  system  of  surfaces  repre- 
sented by  such  an  equation  is  found  in  the  same  manner  as  thai 
of  a  system  of  curves.  See  Diff.  Calc,  Art.  365.  That  is  to 
say,  we  ehminate  the  arbitrary  parameter  from  the  given  equa- 
tion by  means  of  its  derivative  with  respect  to  this  parameter. 

283.  For  example,  in  the  complete  integral  found  in  Art 
281,  if  a  and  ^  are  connected  by  the  relation 

/iCOSa  +  ^sina  +  ^  =  o, (i* 

the  equation  becomes 

0*  +  [(^  —  K)  cos (k-\-{y  —  k)  sin a]^  =  r".      ...  (2) 

Taking  the  derivative  with  respect  to  a,  we  obtain 

\{x  —  K)  cos  a  +  (J^'  —  ^)  sin  a]  [(j;  —  k)  cos  a  —  {x  —  h)  sin  a]  =  o, 

whence  we  must  have  either 

(;c  — //)  cos  a  +  (>' —  ^)  sin  a  =  o,      ....  (3) 
or  else 

(j  —  ^)  cos  a  —  (::»:  —  ^)  sin  a  =  o (4) 

The  elimination  of  a  from  equation  (2)  by  means  of  equation  (3) 
'^ives 

z-^c-, (5) 

and,  in  like  manner,  from  equations  (2)  and  (4)  we  obtain 

z^Jr{x-hy-^  {^y-kf^c- (6) 

Equation  (i)  expresses  the  condition  that  the  axis  of  the  cylin- 
der represented  by  the  complete  integral  shall  pass  through  the 
fixed  point  (//,  k,  o) ;  accordingly  the  envelope  of  the  system  (2) 
consists  of  the  planes  z=±c,  and  the  sphere  (6)  whose  centre  is 


§  X  X 1 1.  ]  EXPRESSION  OF  THE  GENERAL  INTE  GRAL.         3 1 1 

(//,  k,  6).  Regarding  /i  and  k  as  arbitrary,  equation  (6)  is  the 
complete  integral  from  which  as  a  primitive  the  differential 
equation  was  derived  in  Art.  275. 

284.  To  express  the  general  integral,  the  relation  between 
the  constants  in  the  complete  integral  must  be  arbitrary.  Thus, 
the  complete  integral  being  in  the  form 

/(x,y,z,  a,d)=zo, (i) 

we  may  put  d  =  cf}  (a),  where  <^  denotes  an  arbitrary  function,  and 
then  the  general  integral  is  the  result  of  eliminating  a  between 
the  equations, 

/[x,y,z,a,^{a)^  =  o, (2) 

and 

^/[x, y,  z,  a,  cj>(a)^  =  o (3) 

The  elimination  cannot  be  performed  until  the  form  of  <^  is  spe- 
cified ;  for,  as  remarked  in  Art.  275,  the  general  integral  cannot 
be  expressed  by  a  single  equation  unless  the  given  partial  differ- 
ential equation  is  linear. 

Since  the  general  integral  can  thus  be  expressed  by  the  aid 
of  any  complete  integral,  we  shall  hereafter  regard  a  non-linear 
partial  differential  equation  as  solved  when  a  complete  integral 
is  found. 

Singular  Solutions. 

285.  There  may  exist  a  surface  which  at  .each  of  its  points 
touches  one  of  the  surfaces  included  in  the  complete  inte- 
gral without  passing  through  the  corresponding  characteristic. 
Every  element  of  such  a  surface  obviously  satisfies  the  differen- 
tial equation,  and  its  equation,  not  being  included  in  the  general 
integral,  is  a  singiUar  solution  analogous  to  those  which  occur 
in  the  case  of  ordinary  differential  equations. 


312  EQUATIONS  NOT  OF  THE  FIRST  DEGREE.  [Art.  285. 

An  integral  surface  generated,  as  in  Art.  279,  by  a  moving 
characteristic  will  in  general  touch  the  surface  representing  the 
singular  solution  along  a  line.  If  the  surfaces  of  the  complete 
integral  have  this  character,  the  singular  solution  will  be  a  part 
of  the  envelope  found  by  the  process  given  in  the  preceding- 
article,  no  matter  what  the  form  of  <^  may  be.  In  this  case,  equa- 
tions (2)  and  (3),  Art.  284,  which  together  determine  the  ultimate 
intersection  of  consecutive  surfaces  of  the  system  (2),  represent 
a  characteristic  and  also  the  line  of  tangency  with  the  singular 
solution.  The  former,  as  a  varies,  generates  a  surface  belong- 
ing to  the  general  integral,  and  the  latter  generates  the  singular 
solution.  Thus,  in  the  example  of  Art.  283,  equation  (3)  deter- 
mines upon  the  cylinder  (2)  its  lines  of  contact  with  the  planes 
z^±Cy  and  equation  (4)  determines  a  characteristic. 

286.  There  is,  however,  when  a  singular  solution  exists,  a 
special  class  of  integrals  which  touch  the  singular  solution  in 
single  points,  each  of  these  being  in  fact  the  envelope  of  those 
members  of  the  complete  integral  which  pass  through  a  given 
point  on  the  singular  solution.  This  class  of  integrals  obviously 
constitutes  a  doubly  infinite  system,  and  thus  forms  a  complete 
integral  of  a  special  kind.  The  complete  integral  (6),  Art.  283, 
is  an  example. 

When  f{x,y,z,a,b)=o 

is  the  complete  integral  of  this  special  kind,  the  characteristics 
represented  by  equations  (2)  and  (3),  Art.  284,  will,  for  given 
values  of  a  and  b,  all  pass  through  a  common  point,  indepen- 
dently of  the  form  of  <^,  and  this  point  will  be  upon  the  singu- 
lar solution.     In  particular,  the  characteristic  defined  by  /=  o 

and    -f.=zo   will  intersect  that  defined  by  y"=o  and   -Z  =  o, 
aa  db 

in  a  point  on  the  singular  solution.     Hence,  in  this  case,  the 

singular  solution  will  be  the  result  of  eliminating  a  and  b  from 

the  three  equations. 


§  XXII.]  SINGULAR   SOLUTIONS.  313 

^=°'     i=°'      i=°- 

It  is  to  be  noticed,  however,  that  the  eliminant  of  these 
equations  may,  as  in  the  case  of  ordinary  differential  equations, 
include  certain  loci  which  are  not  solutions  of  the  differential 
equation. 

287.  Since  the  characteristics  which  lie  upon  a  surface  of  the 
kind  considered  above,  all  pass  through  the  point  of  contact  with 
the  singular  solution,  it  follows  that  the  singular  solution  is  the 
locus  of  a  point  such  that  all  the  characteristics  which  pass 
through  it  have  a  common  element.  At  such  a  point,  therefore, 
the  initial  element  fails  to  determine  the  direction  of  the  charac- 
teristic. Now,  in  the  equations  (10),  Art.  277,  the  ratio  dx'.dy 
is  indeterminate  only  when  P  —  o  and  Q  =  o,  or  when  P  =  00 
and  G  =  00  ;  hence  one  of  these  conditions  must  hold  at  every 
point  of  a  singular  solution.  The  former  is  the  more  usual  case, 
so  that  a  singular  solution  generally  results  from  the  elimination 
of/  and  q  from 

F{x,y,z,p,q)  =  o 

by  means  of  the  equations 

dF  .  dF 

=  o  and  =  o. 

d^  dq 

It  is  necessary,  however,  to  ascertain  whether  the  locus  thus 
found  is  a  solution  of  the  differential  equation,  for  the  conditions 
P  =  o,  Q  =  o,  and  P=cc,  Q=oo  are  satisfied  at  certain  other 
points  besides  those  situated  upon  a  singular  solution  ;  for  ex- 
ample, those  at  which  all  the  characteristics  which  pass  through 
them  touch  one  another.  In  the  example  of  Art.  278,  P  =  o, 
Q  =  o  gives  the  singular  solution  ^  =  ±  r,  and  P=oo,  Q  ='00 
gives  2^  =  o,  which  is  the  locus  of  the  last-mentioned  points, 
and  not  a  solution. 


314         EQUATIONS  NOT  OF  THE  FIRST  DEGREE.  [Art.  288. 


Equations  Involving  p  and  q  only, 

288.  We  proceed  to  consider  certain  cases  in  which  a  com- 
plete integral  is  readily  obtained.  In  the  first  place,  let  the 
equation  be  of  the  form 

^(A^)  =  o (i) 

In  this  case,  since  X=o,  F=o,  Z=o,  two  of  the  equations 
[(10),  Art.  277]  of  the  characteristic  become  dp  =  o  and  dq  =  o\ 
whence 

p  ■=i  a  and  q  ■=^  b (2) 

The  constants  a  and  b  are  not  independent,  for,  substituting  in 
equation  (i),  we  have 

^(^,  ^)  =  o (3) 

Substituting  in  dz  z=pdx  -\-  qdy,  we  obtain 

dz  =  adx  4"  bdy ; 
whence,  integrating,  we  have  the  complete  integral 

z  =  ax  +  by  -j-  c, (4) 

where  a  and  b  are  connected  by  equation  (3),  and  r  is  a  second 
arbitrary  constant. 

289.  The  characteristics  in  this  case  are  straight  lines,  and 
the  complete  integral  (4)  represents  a  system  of  planes.  The 
general  integral  is  a  developable  surface.  There  is  no  singular 
solution. 

A  special  class  of  integrals  which  may  be  noticed  are  the 
envelopes  of  those  planes  belonging  to  the  system  (4)  which 
pass  through  a  fixed  point.*     These  are  obviously  cones,  whose 

*  The  characteristics  which  pass  through  a  common  point  in  all  cases  determine 
an  integral  surface.  The  integrals  of  this  special  kind  constitute  a  triply  infinite 
system :  we  may  limit  the  common  point  or  vertex  to  a  fixed  surface  (as,  for  example, 
in  Art.  286,  to  the  singular  solution),  and  still  have  a  complete  integral. 


§  XXII.]  EQUATIONS  ANALOGOUS    TO   CLAIRAUT'S.  315 

elements  are  the  characteristics  which  pass  through  the  fixed 
point.     For  example,  if  the  equation  is 

^2  -f-  ^2  =  ffi'^^ 

these  cones  are  right  circular  cones  with  vertical  axes,  and  their 
equations  are 

Equation  Analogous  to  Clairaufs. 

290.  There  is  another  case  in  which  the  characteristics  are 
straight  lines  ;  namely,  when  the  equation  is  of  the  form 

z=px  +  qy+f{p,q) (i) 

In  this  case,  X=p,  V=  q,  Z=—i,  and  we  have  again,  for  two 
of  the  equations  of  the  characteristic,  dp  =  0  and  dq  —  Q\  whence 

p=a,  q^b (2) 

Substituting  in  dz  =pdx  +  qdy,  and  integrating,  we  have  the 
complete  integral 

z  =  ax  +  by  -\-  c, (3) 

in  which  the  constant  c  is  not  independent  of  a  and  b  ;  for,  sub- 
stituting the  values  of  p  and  q,  equation  (i)  becomes 

z  =  ax-\-  by  -\-f{a,  b), (4) 

which,  since  it  is  also  one  of  the  integrals  of  the  characteristic 
system,  must  be  identical  with  equation  (3). 

291.  The  complete  integral  in  this  case  also  represents  a 
system  of  planes,  and  the  general  integral  is  a  developable  sur- 
face.    A  singular  solution  also  exists. 


3l6         EQUATIONS  NOT  OF  THE  FIRST  DEGREE.        [Art.  29 1. 

For  example,  let  the  equation  be 

z=^px-^qy^-ksj{\-\-p''-\-q^)', (i) 

the  complete  integral  is 

z— ax^-by-^rksJ^Y-^-a^ -\-b^') (2) 

For  the  singular  solution,  taking  the  derivatives  with  respect  to 
a  and  by  we  have 


and 


bk 
^  ^  siix  ■\- a- ■\- b-) 


These  equations  give 


3  = 


-y 


~  \j  {k""  —  x"" — y^y        ~  v/(^2  _  ^2  _^2) 

and,  substituting  in  equation  (2),  we  have 

x^  +  y^ -\- z""  =  k^ (3) 

Thus  the  singular  solution  represents  a  sphere,  the  complete 
integral  (2)  its  tangent  planes,  and  the  general  integral  the 
developable  surface  which  touches  the  sphere  along  any  arbi- 
trary curve. 

Equations  not  Containing  x  or  y. 

292.    When  the  independent  variables  do  not  explicitly  occur, 
the  equation  is  of  the  form 

F{z,p,q)  =  o (i) 


§  XXII.]  EQUATIONS   OF  SPECIAL   FORMS.  317 

Here  X—O  and  F=o,  and  the  final  equation  of  the  character- 
istic system  reduces  to 

dp       dq 

whence 

q  =  ap (2) 

Substituting  in  equation  (i),  we  have  F{z,p,ap)  =0,  the  solu- 
tion of  which  gives  for/  a  value  of  the  form 

/  =  </>(2). 
Thus,  dz  —pdx  +  qdy  becomes 

dz  —  (f){z)  (dx  4-  ady)  f 
whence,  integrating,  we  have  the  complete  integral, 

X  -^  ay  = \-  ^ (3) 

The  illustrative  example  of  Arts.  278  and  281  is  an  instance 
of  this  form.  It  will  be  noticed  that  the  mode  of  solution  leads 
to  a  complete  integral  representing  cylindrical  surfaces  whose 
elements  are  parallel  to  the  plane  of  jry.     The  equation 

F{z,  o,  o)  =  o, 

representing  certain  planes  parallel  to  the  plane  of  xy,  will  obvi- 
ously be  the  singular  solution. 

Equations  of  the  Form  f^{x, p)=f2{y,  q). 

293.  When  the  equation  does  not  explicitly  contain  z,  it  may 
be  possible  to  separate  the  variables  x  and  /  from  y  and  q,  thus 
putting  the  equation  in  the  form 

Mx,p)^My,q) (i) 


3l8        EQUATIONS  NOT   OF  THE  FIRST  DEGREE.        [Art.  293. 

In  this  case,  we  have  Z=  o,  X=^  -4^,  P=  -4^,  and  the  equations 

dx  dp 

of  the  characteristic  give  for  the  relation  between  dx  and  dp^ 

Integrating,  we  have/i(;ir,/)  :»^,  and  from  equation  (i), 

A{x,p)=My,q)  =  a (2) 

Solving  these  equations  for/  and  q,  we  have  values  of  the  form 

p  =  <f>,{x,a),  ^==My,^)> 

and  d2  =pdx  +  gdy  becomes 

dz  =  <f>^(x,  d)dx  +  <^2(7,  d)dy, 
whence  we  derive  the  complete  integral, 

z  =  Ui(^,  d)dx  +  Ui2{y,  d)dy  +  b. 

For  example,  let  the  given  equation  be 

xp-^-^-yq"^  =  I. 
Putting 

xp"^  =  I  —  yq"^  =  a, 

we  have 

and,  integrating  ds  =pdx  +  qdy,  we  obtain  the  complete  integral, 
z  =  2sja^x  +  2  sj{i  —  a)\Jy  +  b. 


§  XXIL]     CHANGE  IN  THE  CHARACTERISTIC  EQUATIONS.    319 


Change  of  Form  in  the  Equations  of  the  Characteristic. 

294.    If  we  make  any  algebraic  change  in  the  form  of  the 
equation 

F{x,y,z,p,  q)  =  o, 

the  equations  of  the  characteristic  (10),  Art.  277,  will  be  altered. 
The  changes,  however,  will  be  merely  such  modifications  as 
might  be  produced  by  means  of  the  equation  F=o  itself.*  In 
particular,  the  form  assumed  when  the  equation  is  first  solved 
for  g  may  be  noticed.     Suppose  the  equation  to  be 

q  =  <i>{x,  y,  z, p) (i) 

Then  F=^-«^(;r,j,^,/),  whence  X=-^,  F=-^,  Z  =  -^, 

dx  dy  dz 

P  — 5^,  and  0=1.     Putting  q  in  the  place  of  ^  in  the  partial 

dp 

derivatives,  and  omitting  the  member  containing  dq,  the  equa- 
tions of  the  characteristic  become 

J^^dy^       ^^      = ±—,     .    .    .    .  (2) 

dp  ^      ^  dp      dx^^dz 

a  complete  system  for  the  four  variables  x,  j/,  z  and  /,  q  being 
the  function  of  these  variables,  given  by  equation  (i).  These 
equations  may  be  deduced  from  the  consideration  that  the  val- 
ues of/  and  q  derived  from  one  of  their  integrals  combined  with 
equation  (i)  should  render  dz=pdx  -\- qdy  integrable.f 


*  The  complete  solution  of  the  characteristic  system  involving  four  arbitrary 
constants  (see  Art.  278)  would  indeed  be  changed,  but  not  the  special  solution  in 
which  7^=  o  is  taken  as  one  of  the  integrals. 

t  See  Boole's  "Differential  Equations,"  London,  1865,  p.  336. 


320        EQUATIONS  NOT   OF  THE  FIRST  DEGREE.        [Art.  295. 

295.   As  an  illustration,  let  us  take  the  equation 

z^pq,  or  q=- (l) 

Equations  (2)  of  the  preceding  article  become 

p'^dx        ,       pdz      j^  ,  . 

Z  2Z 

Of  these  the  most  obvious  integral  is 

p=y-\-  a; 

whence  ds  =^pdx  -f  qdy  becomes 

dz^{y^-d)dx  +  -^, 

from  which  we  derive  the  complete  integral 

z={y^d){x-Vb) (3) 

The  equations  of  the  characteristic  derived  from  the  more 
symmetrical  form  of  the  equation 

F  =■  pq  —  z  —  o 
are 

dx  __dy  _  dz   _dp  _dq  ^  ^ 

q  '~  p  "  2pq~  p  "  q' 

which  are  readily  seen  to  be  equivalent  to  equations  (2).  If  the 
final  equation  of  the  system  (4)  be  used,  as  in  the  process  of 
Art.  292,  to  determine/  and  q,  we  shall  have 

/=— ,  q  =  a^Z, 

giving 

4z==(^  +  ay-\-^J, (5) 

another  complete  integral  of  the  equation  ^  =/^. 


§  XXII.]  TRANSFORMATION  OF   THE    VARIABLES.  32 1 


Transfor7nation  of  the  Variables. 

296.  A  partial  differential  equation  may  sometimes  be  re- 
duced by  transformation  of  the  variables  to  one  of  the  forms  for 
which  complete  integrals  have  been  given  in  Arts.  288,  290,  292 
and  293.  The  simplest  transformation  is  that  in  which  each 
variable  is  replaced  by  an  assumed  function  of  itself.  The 
choice  of  the  new  variable  will  be  suggested  by  the  form  of 
the  given  equation. 

Let 

then 


^^-fi^y^-/'i^)(f/i^f/v) 


Hence,   denoting   the  partial  derivatives  of  t  with  respect  to 
^  and  r]  by  /'  and  q',  their  expressions  in  terms  of  x,j/  and  ^  are 
the  same  as  if  they  were  ordinary  derivatives. 
For  example,  the  equation 

may  be  written 

\zdxj       \zdy) 

Putting  —  =  d^,  -^=  df),  ~  =  dt,  whence  ^  =  log;r,   17  =  logy 

X  y  z 

and  ^  =  log  ^,  the  equation  becomes 

/^  -f-  /^  =  I (2) 

The  complete  integral  of  this  equation  is,  by  Art.  288, 
1  =  ai-\-  by]-\-  c, 


322       EQUATIONS  NOT  OF  THE  FIRST  DEGREE.        [Art.  296. 

•where  a"^  -^  b""  =.  i\  hence,  putting  a  =  cos  o.y  b  —  sin  a,  the  com- 
plete integral  of  equation  (i)  is 

log  2  =  cos  a  log  ^  +  sin  a  log^  +  c^ 
or 

297.   In  the  following  example  the  new  independent  variables 
are  functions  of  both  of  the  old  ones.     Given 

{x-^y-){p--\-g-)  =  1 (i) 

Using  the  formulae  connecting  rectangular  with  polar  coordi- 
nates, 

x  —  r  cos  B,  y  —  r  sin  0, 

whence 
we  have 


y 

r^z=x'-^y^,  ^  =  tan-^-, 


dz       dz        /)  dz  sin^ 

p-=.  —  =  —  cos  d , 

^       dx      dr  dB      r 

dz       dz    .    ^  .  dz  cosB 

^       dy       dr  dB      r 

Substituting,  equation  (i)  becomes 

'dz 


or,  putting  dp  =  — , 


*-M 


Hence  the  integral  is 

z  —  p  cos  a  +  ^  sin  a  +  /8 

y 

=  \  cos  a  log  {x"^  +JV^)  +  sina  tan-^-  +  ^^ 

The  same  complete  integral  may  be  found  directly  by  the  method 
of  characteristics  (see  Ex.  20). 


§  XXIL]  EXAMPLES.  323 

Examples  XXIL 

Find  complete  integrals  for  the  following  partial  differential 
equations  :  — 

y 

i.pq=i,  z  =  ax-\--  +  b. 

2.  \lp  -\-  sjq—  2Xy  ^  =  i  (2^  —  aY  +  CL'^y  +  b. 

3.  /2  _  ^2  _  j^  2  =  :r  sec  a  -\-  y  tana  +  ^. 

I.   z  =  px  +  qy  -{-  J>q,  z  =  ax  -{-  by  +  ab  ; 

singular  solution,  z  —  —  xy. 

S-  q  —  xp  +  p^,  z  =  axey  +  I  o'e'^y  +  b. 

6.  y^p^  —  x^q^  =  x^y^,  z  =  ^  ax""  -j-  ^  {a^  —  i)^;'^  +  b. 

7'  P'  +  r  =  ^+y,  z  =  i{x-\-ay--{-^{y-ay-  +  b. 

8.  q  =  ^yp"",  z  =  ax  -\-  a'^y''  +  b. 

L    l  I    £ 

9.  2  =  /jj;  +  ^j^  --  np*^q**,  z  =  ax-\-by~  na'^b^  ; 

singular  solution,  z  —  (2  —  n)  {xyY~\ 

10.  p^  -  q^  =  x^2Z^  z^  ^  rr  -r  ^)^  +  (7  +  ^)''  +  ^. 

z 

11.  p  =  (qy -\- zy,  yz  —  ax-\-2sl{ay)+b. 

12.  /2  _j_  ^2  _  2^x  —  2qy  +1  =  0, 

2Z   =    ;(:2    _|_jj;2    _|_  ^^(jt:2    _j_    ^)    J^  y^(^y2   _    j    _    ^) 

13.  Denoting  x -\- ay  by  /,  find  a  complete  integral  of  Ex.  12  in 
the  form 

2(iH-a=)2  =  /2  +  /y/(/2_i_a2)_(i4.^2)  iog[/+y/(/2_i-.a»)]  +  /J. 

14.  (/  +  ^)(/^  +  ^i')  =  I,  v^(i +^)2=  2v'(^  ■h^jJ') +^. 


324        EQUATIONS  NOT   OF  THE  FIRST  DEGREE.         [Art.  297. 


15.  x-y^z^pq-  =1,  ^22  =  _  _i_  _  1^  _^  ^, 

a^x       sjy 

16.  x-^y^z'^p-'q  =  I,  log—  =  — ^. 

bz       2  ay 

17.  p2-y2^^^y:,^^2^ 

z  =  —  sin-»  -  H — 5lj^ i V  +  ^. 

2         a  2  y 

18.  Find  three  complete  integrals  of    . 

pq=px-^qy. 

1°   22  =  (^^  +  a>'Y4- y8. 

2°   2;  =  xy  +  7  ^(^2  —  a"^^   +  <^. 

19.  Show  directly,  by  comparison  of  the  values  of  2,  /  and  q,  that  a 
surface  included  :^  the  integral  2°  can  be  found  touching  at  any  given 
point  a  given  surface  mrluded  in  the  integral  1° ;  and  that  the  relation 

^^2b-^- 

will  then  exist  between  the  constants.  Hence  derive  one  integral  from 
the  other,  as  in  Art.  283.  Also  show  that  the  similar  relations  for  the 
other  pairs  of  integrals  are 

y8  =  2^'  4-  «'^a2,  and  b  —  d^  =  aa\ 

20.  Show  that  xq  —  yp=  a  is  an  integral  of  the  characteristic  system 

for  the  equation 

{x-+y-){p-  +  q^)  =  i; 

and  thence  derive  the  complete  integral  given  in  Art.  297. 

2 1 .  Solve,  by  means  of  the  transformations  xy  =.  ^,  x  -\-y  =  v-  the 
equation 

(y  -  ^')  {qy  -P^)  =  {P-  qY- 

z  =  axy  +  C^i^x  +7)'+  b. 

22.  (a:«  —  y^^pq  —  xy{p^  —  q"")  =  i- 

s  =  ^a\og{x^  +jO  H--tan-^^+<5. 


§XXII.]  EXAMPLES.  325 

23.  Show  that  the  equations  of  the  characteristic  passing  through 
(a,  )8,  y)  in  the  case  of  the  equation 

Art.  289,  are 

X  —  a  _  y  —  p  _Z  —  y 
a  b  m^ 

where  a^  -\-  b""  =  m"^  -,   and  thence  derive  the  special  integral  given  in 
that  article. 

24.  Deduce,  in  like  manner,  the  integral  formed  by  characteristics 
passing  through  {h,  k,  I)  for  the  equation 

^^  +  ^^  =  --1. 

(^  _  hy  +{y-  ky  =  lsj{c-  -  l^)-sj{c-  -  z-)Y. 

25.  Show  that  when  the  complete  integral  is  of  the  form 

au  +  bv -^  w  =  o, (i) 

where  u,  v  and  w  are  rational  functions  of  x,  y  and  z,  the  elimination 
can  be  performed,  giving  the  general  integral 


<^ 


f^,   ^)  =  o, (2) 


a  homogeneous  equation  in  u,  v,  w.  Accordingly,  show  that  the  equa- 
tion arising  from  equation  (i)  as  a  primitive  is  the  linear  equation 
Pp  -\-  Qq  =  R,  where 

d{y,  2)        ^{y,  2)         d{y,  z) 

U  7) 

with  similar  expressions  for  Q  and  R,  and  that  putting  u^  —  —',    v^  =—, 

w  lu. 

these  values  of  P,  Q  and  R  agree  with  those  derived  from  the  general 
primitive  in  Art.  271. 


326  EQUATIONS   OF  THE   SECOND   ORDER.  [Aft.  298. 


CHAPTER   XII. 

PARTIAL   DIFFERENTIAL   EQUATIONS    OF    HIGHER   ORDER. 

XXIII.  . 

Equations  of  the  Second  Order, 

298.  We  have  seen  that  the  general  solution  of  a  partial 
differential  equation  of  the  first  order,  containing  two  independ- 
ent variables,  involves  an  arbitrary  function,  although  it  is  not 
possible  to  express  the  solution  by  a  single  equation  except 
when  the  differential  equation  is  linear  with  respect  to  /  and  q. 
We  might  thus  be  led  to  expect  that  the  general  solution  of  an 
equation  of  the  second  order  could  be  made  to  depend  upon  two 
arbitrary  functions.  But  this  is  not  generally  the  case.  No 
complete  theory  of  the  nature  of  a  solution  has  yet  been  devel- 
oped, although  in  certain  cases  the  general  solution  is  expressi- 
ble by  an  equation  containing  two  arbitrary  functions.  We  shall 
consider  these  cases  in  the  present  section,  and  in  the  next,  the 
important  class  of  linear  equations  with  constant  coefficients, 
for  which  in  some  cases  a  solution  of  the  equation  of  the  ;zth 
order  containing  n  arbitrary  functions  can  be  obtained. 

The  Primitive  containing  Two  Arbitrary  Functions. 

299.  If  we  consider  on  the  other  hand  the  question  of  the 
differential  equation  arising  from  a  given  primitive  by  the  elimi- 
nation of  two  arbitrary  functions,  we  shall  find  that  it  is  only  in 


§  XXIII. ]  TJVO  ARBITRARY  FUNCTIONS.  527 

certain  cases  that  the  eUmination  can  be  performed  without 
introducing  derivatives  of  an  order  higher  than  the  second. 

The  general  equation  containing  two  arbitrary  functions  may 
be  written  in  the  form 

f{x,y,z,  cfi(u),  ilf(v)^  =  o, 

in  which  u  and  v  are  given  functions  of  x,  y  and  z.     The  two 

derived  equations 

df  df 

^^  =  o,  -^  =  o, 

dx  dy 

will  contain  4>\ti)  and  ^'{v),  two  new  unknown  quantities  to  be 
eliminated.  There  will  be  three  derived  equations  of  the  second 
order 

^=0,      ^^=0,      ^=0, 

dx^        '  dxdy        '  dy"" 

containing  two  new  unknown  quantities,  <^"(?/)  and  «/'"(^)-  We 
have  thus  in  all  six  equations  containing  six  unknown  quantities. 
The  elimination,  therefore,  cannot  in  general  be  effected.* 

300.  Suppose,  however,  that  the  original  equation  can  be  put 
in  the  form 

w  =  <f>{u)  +ip(v); (i) 

then  the  two  derived  equations  of  the  first  order, 

dw   ,   dw  .        ,,.   .(du   ,   du  \   ,    ,,.  ^(dv    ,   dv  \  ,  s 

- + -/ = ,^  («)  (^- + _/j  +  f  (.)  ^_ + -py .  .  (.) 

dw       dw  iu..\(du   .   du    \   ,    ,,.  ^(dv    .   dv    \  ,   \ 

are  independent  of  <^  and  \\i.     These,  with   the   three  derived 

*  If  we  proceed  to  the  third  derivatives,  we  shall  have  ten  equations  and  eight 
quantities  to  be  eliminated,  so  that  two  equations  of  the  third  order  could  be  found 
which  would  be  satisfied  by  the  given  primitive. 


328  EQUATIONS   OF  THE  SECOND    ORDER.  [Art.  3CX). 

equations  of  the  second  order,  will  constitute  five  equations 
containing  the  four  quantities  <i>\  \j/',  <^",  ^A"-  These  quantities 
may  therefore  be  eliminated,  the  result  being  an  equation  of  the 
second  order. 

There  is  another  way  in  which  the  elimination  may  be 
effected.  Let  one  of  the  unknown  quantities,  say  i/^',  be  elimi- 
nated between  equations  (2)  and  (3) ;  we  shall  then  have  a  single 
equation  containing  <i>'.  From  this  equation  and  its  two  derived 
equations  we  can  eliminate  <^'  and  <^".  It  is  to  be  noticed  that 
in  this  last  process  we  meet  with  an  intermediate  equation  of  the 
first  order,  containing  one  arbitrary  function. 

301.  Another  case  in  which  the  elimination  can  be  per- 
formed occurs  when  the  primitive  is  of  the  form 

W=  (ji{u)  -^  zf\p{u), (i) 

in  which  we  have  two  arbitrary  functions  of  the  same  given 
function  of  x,  y  and  s.  In  this  case  the  derived  equations  take 
the  form 

in  which  f  — ),  etc.,  are  written  in  place  of  — +-^/,  etc. 
\dx  J  dx       dz 

Multiplying  equations  (2)  and  (3)  by  [  —  jand  (— j  respectively, 

and  subtracting  the  results,  <^'(?/)  and  »/''(//)  are  eliminated  to- 
gether, and  we  have  again  an  intermediate  equation  of  the  first 
order  containing  one  arbitrary  function.* 

*  The  cases  considered  in  this  and  the  preceding  article  are  not  the  only  ones 
in  which  an  intermediate  equation  of  the  first  order  can  arise.  See,  for  instance,  the 
example  given  in  Art.  311. 


§  XXIIL]  THE  INTERMEDIATE  EQUATION.  329 


The  Intermediate  Equation  of  the  First  Order. 

302.  The  preceding  articles  indicate  two  cases  in  which  an 
intermediate  equation  of  the  first  order  may  arise  from  a  primi- 
tive. We  have  now  to  consider,  on  the  other  hand,  the  form  of 
the  differential  equations  arising  from  an  intermediate  equation 
of  the  form 

u  =  ^(v), (i) 

where  u  and  v  now  denote  given  functions  of  x,  f,  Zy  p  and  q* 
and  <^  is  an  arbitrary  function.  Denoting  the  second  derivatives 
of  ^  by  r,  J  and  /,  thus 


r  = 

d-z 
dx^"* 

d^z 
dxdy 

the  two  derived 

equations 

are 

l^f^^ 

dp 

dq 

=  *'(^)(l 

^  dz^      dp 

dv 
1q 


du       du         du      \^^f—AJ(\  f^"^'   i^'^     4_  ^    A-—t 
dy       dz  dp         dq  \dy       dz  dp         dq 


and  the  result  of  eliminating  <fi\v)  may  be  written  in  the  form 

=  0.    (2) 
Of   the  sixteen  determinants  formed   by  the  partial   columns, 

*  In  the  cases  considered  in  Arts.  300  and  301,  the  function  v  in  the  intermediate 
equation  does  not  contain  /  and  q,  but  we  here  include  all  cases  in  which  an  inter- 
mediate equation  of  the  first  order  can  exist. 


du    ,    .du   ,      du    ,      du  du   ,      du   ,      du   .     .du 

\-p [- r \- s — \- q \- s \- t  — 

dx         dz  dp         dq  dy         dz  dp         dq 

dv   .    .dv   .      dv   .      dv  dv    .      dv  ,      dv    ,    .dv 

\- p \- r \- s — \- q \- s \- t  — 

dx         dz  dp  dq  dy  dz  dp  dq 


330 


EQUATIONS   OF  THE  SECOND    ORDER.  [Art.  302. 


those  containing  rs  and  st  vanish,  and  the  remaining  terms  of 
the  second  degree  in  r,  j,  /  may  be  written 


(r/-^0 


dp 

du 
dq 

dv 

dv 

dp 

dq 

or         {rt  —  s^) 


d{}i,  v) 


The  equation  will  therefore  be  of  the  form 

Rr-^Ss  +  Tt-\-  U{rt  -  s^)  =  V. 


(3) 


303.  When  a  given  equation  of  the  second  order  admits  of  a 
primitive  containing  two  arbitrary  functions,  the  intermediate 
equation  of  the  first  order  is  an  mtermediate  integral  analogous 
to  the  first  integrals  of  ordinary  differential  equations  of  the 
second  order.  It  follows  from  the  preceding  article  that  an 
intermediate  integral  will  exist  only  when  the  given  equation  is 
of  the  form  (3),  and  when  ?/  and  v  can  be  so  determined  as  to 
make  the  functions  R,  S,  7",  U  and  F  defined  by  the  develop- 
ment of  equation  (2)  proportional  to  the  given  coefificients.  This 
imposes  four  conditions  upon  the  two  quantities  ii  and  v ;  hence 
two  identical  relations  must  exist  between  the  coefficients,  in 
order  that  an  intermediate  integral  may  be  possible. 


Successive  Integration. 

304.  When  an  intermediate  integral  can  be  found,  the  final 
integral  is  derived  from  it  by  the  methods  given  in  the  preceding 
chapter  for  equations  of  the  first  order ;  the  second  integration 
introducing  a  second  arbitrary  function. 

In  some  simple  cases  it  is  obvious  that  an  intermediate 
equation  can  be  obtained  by  direct  integration.  Thus,  when  the 
equation  contains  derivatives  with  respect  to  one  only  of  the 


§  XXI 1 1.]  SUCCESSIVE  INTEGRATION.  33 1 

independent  variables,  we  may  treat  it  as  an  ordinary  differential 
equation  of  the  second  order ;  taking  care  only  to  introduce  arbi- 
trary functions  of  the  other  variable  in  the  place  of  constants  of 
integration.     Given,  for  example,  the  equation 

xr  —  p  ■=■  xy, 

which  may  be  written 

xdp  —  p^x  _  ydx 


Integrating, 


X'^  X 

p 

-=y\o%x-ir^{y). 


or 


p  =  yx\ogx -^  Xi^{y)', 

and,  integrating  again, 

z^yi^^x-^Xogx-lx^)  -^rhx-'i^^y)  -f  «AW> 

or,  putting  ^{y)  in  place  of  the  function  \^{y)  —  \y, 

z  =  ^yx''  \ogx  +  ^^<^(j)  +  ^{y). 

305.  Again,  an  equation  which  does  not  contain  t  may  be 
exact  *  with  reference  to  x,  y  being  regarded  as  constant.  Given, 
for  example,  the  equation 

/  +  r  + J  =  I  j 
integrating,  we  have 

*  The  equation  might  also  be  such  as  to  become  exact  with  respect  to  the  four 
variables  p,  q,  z  and  x,  by  means  of  a  factor.  For  this  purpose  three  conditions  of 
integrability  would  have  to  be  satisfied;  see  Art.  252.  This  is  the  number  of  con- 
ditions we  should  expect,  since  by  Art.  303  two  must  be  fulfilled  to  render  an  inter- 
mediate integral  possible,  and  one  more  is  necessary  to  express  that  in  that  integral 
v=  y. 


332  EQUATIONS   OF  THE  SECOND   ORDER.  [Art.  305. 

For  this  linear  equation  of  the  first  order,  Lagrange's  equations 
are 

dx^dy^ ^^ , 

of  which  the  first  gives 

x-y=a, 

and  this  converts  the  second  into 

dy 
of  which  the  integral  is 

evz  =  aey  +  \Xy  +  </)(j)]<?^^  +  h. 
Hence,  making  b  =  ^{a)y  we  have  for  the  final  integral 
eyz  =  eVx  -eyy^-[\_y-^  <^  W]  ^^^JV  +  ^{x  -y), 
or,  with  a  change  in  the  meaning  of  <^, 

z  —  X  -\-  <f>(y)  -\-  e-y\p{x  —7). 

Mongers  Method. 

306.  The  general  method  of  deriving  an  intermediate  equa- 
tion where  one  exists  is  based  upon  a  mode  of  reasoning  similar 
to  the  following  method  for  Lagrange's  solution  of  equations  of 
the  first  order,  which  is  that  by  which  it  was  originally  estab- 
lished. 

Given  the  equation 

Fp  +  Qq  =  R, (i) 

and  the  differential  relation 

dz  =  pdx -\- qdy, (2) 


§  XXIIL]  MONGERS  METHOD.  333 

which  must  exist  when  ^  is  a  function  of  x  and  y.  Let  one  of 
the  variables  p  and  q  be  eUminated,  thus 

dy 
or 

p{Pdy  -  Qdx)  +  Qdz  -  Rdy  =  o (3) 

Hence,  the  relation  between  x,  y  and  z  which  satisfies  equation 
(i)  must  be  such  that,  when  one  of  the  two  differential  expres- 
sions occurring  in  equation  (3)  vanishes,  the  other  will  in  general 
also  vanish.     Let  us  now  write  the  equations 


Pdy  -  Qdx 
Qdz  -  Pdy 


:}• ■<" 


and  suppose  u  =  a,  v  =  b,to  he.  two  integrals  of  these  simulta- 
neous equations.  Then  dti  =  o  and  dv  =  o  constitute  an  equiva- 
lent differential  system,  and  the  relation  between  x,  y  and  s  is 
such  that,  if  dti  =  o,  then  dv  =  o\  that  is,  if  ti  is  constant,  v  is  also 
constant.     This  condition  is  satisfied  by  putting 

u  =  <^{v), 

which  is  therefore  the  solution  of  equation  (i). 

Geometrically  the  reasoning  may  be  stated  thus  :  If  upon  a 
surface  satisfying  equation  (i)  a  point  moves  in  such  a  way  that 
Pdy  —  Qdx  =  o,  then  also  will  Qdz  —  Rdy  =  o  ;  that  is,  the  point 
will  move  in  one  of  the  lines  determined  by  equations  (4).  No 
restriction  is  imposed  upon  the  surface,  except  that  it  shall  pass 
through  these  lines,  namely,  Lagrange's  lines  defined  by  ?/  =  a, 
V  =  b.  The  general  equation  of  the  surface  so  restricted  is 
u  —  <t>(v). 

307.    Monge  applied  the  same  reasoning  to  the  equation 

Rr  +  Ss  -h  n=F,     .......   (i) 


334  EQUATIONS  OF  THE  SECOND   ORDER.  [Art.  307. 

where  R^  S,  T  and  F'are  functions  of  x,  y,  z,  p  and  q,  in  connec- 
tion with  which  we  have,  for  the  total  differentials  of/  and  q, 

dp  =  rdx  +  sdyy (2) 

dq  =  sdx  +  tdy (3) 

Eliminating  two  of  the  three  variables  r,  s,  t,  we  have 

j^dp-  sdy  ^^^_^  ^dq  -  sdx  ^  y^ 
dx  dy 

or 

Rdpdy  +  Tdqdx  —  Vdxdy  =  s{Rdy^  —  Sdxdy  +  Tdx'').    .    .   (4) 

If,  then,  we  can  find  a  relation  between  ;r,  y,  z,  p  and  q,  such 
that,  when  one  of  the  two  differential  expressions  contained  in 
equation  (4)  vanishes,  the  other  will  vanish  also,  this  relation 
will  satisfy  equation  (i). 

Let  us  now  write  the  equations 

Rdy""  —  Sdydx  +  Tdx""  =  o  "| 
Rdpdy  +  Tdqdx  =  Vdxdy  J 

If  //  =  <2  and  v  =  b  are  two  integrals  of  this  system,  so  that  dti  =  o, 
and  dv  —  o  form  an  equivalent  differential  system,  the  required 
relation  will  be  such  that  if  du  =  o,  then  dv  =  o;  that  is,  if  u  is 
constant,  v  is  also  constant.  As  in  the  preceding  article  this 
condition  is  fulfilled  by 

^  u  =  <f>(v), 

which  is  now  a  differential  equation  of  the  first  order.  The 
integral  of  this  equation  is  therefore  a  solution  of  equation  (i).* 


*  The  same  method  applies  to  the  more  general  form  (3),  Art.  302,  when  an 
intermediate  integral  exists,  but  the  auxiliary  equations  are  more  complex.  See 
Forsyth's  Differential  Equations,  p.  359  ei  seq. 


§  X  X 1 1 1.  ]      INTE  GRABILIT  V  OF  MONGERS  EQUA  TIONS.  335 

308.  The  auxiliary  equations  (5)  are  known  as  Monges  equa- 
tions.  The  first  is  a  quadratic  for  the  ratio  dy :  dx,  and  is  there- 
fore decomposable  into  two  equations  of  the  form  dy  =  mdx. 
Employing  either  of  these  the  second  equation  becomes  a  rela- 
tion between  dp,  dq  and  dx  or  dy.  These  two  equations,  taken 
in  connection  with 

dz  =  pdx  +  qdy, 

form  a  system  of  three  ordinary  differential  equations  between 
the  five  variables  x,  y,  z,  p  and  q.  Since  four  equations  are 
needed  to  form  a  determinate  system  for  five  variables,  it  is 
only  when  a  certain  condition  is  fulfilled  that  it  is  possible  to 
obtain  by  the  combination  of  these  three  equations  an  exact 
equation  giving  an  integral  u  =  a.  Again,  a  second  condition  of 
integrability  *  must  be  fulfilled  in  order  that  the  second  integral 
V  =.b  shall  be  possible.  These  two  conditions  are  in  fact  the 
same  as  those  mentioned  in  Art.  303,  as  necessary  to  the  exist- 
ence of  an  intermediate  integral  containing  an  arbitrary  function. 

309.  If  R,  S  and  T  in  the  given  equation  contain  x  and  y 
only,  the  first  of  Monge's  equations  is  integrable  of  itself.  Given, 
for  example,  the  equation 

ocr-{x^y)s^yt^y^y{p-^q) (i) 

Monge's  equations  are 

xdy^ -\- {x -\-y^dydx -\-ydx'^  =  Q, (2) 

xdpdy-\-ydqdx^^^^^^{p-q)dydx (3) 

*  When  there  is  a  deficiency  of  one  equation  in  a  system,  a  single  condition 
must  be  satisfied  to  make  an  integral  possible,  just  as  a  single  condition  is  necessary 
when  one  equation  is  given  between  three  variables.  Supposing  one  integral  found, 
one  of  the  variables  can  be  completely  eliminated;  there  is  still  a  deficiency  of  one 
equation  in  the  reduced  system,  and  again  a  condition  must  be  fulfilled  to  make  a 
second  integral  possible. 


336  EQUATIONS   OF  THE  SECOND   ORDER.  [Art.  309. 

Equation  (2)  may  be  written 

{dy  -f-  dx)  {xdy  -\-  ydx)  =  0.  *. 

Taking  the  second  factor,  we  have 

xdy  -\-  ydx  —  o, 

which  gives  the  integral 

xy^a, (4) 

and  converts  equation  (3)  into 

dp  —  dq  _dx  —  dy 
p-q    "    x-y  ' 

This  gives  for  the  second  integral 

^-^=t (5) 

X  —  y  ^-^^ 

Hence  we  have  for  the  intermediate  integral 

^y='i-{xy) (6) 

To  solve  this  equation  of  the  first  order,  Lagrange's  equations 
are 

dxz=^dy= , (7) 

-"       {x-y)<i>{xyy  ^7^ 

of  which  the  first  gives 

x+y  =  a (8) 

For  the  second  integral  we  readily  obtain  from  equations  (7) 

xdx -{- ydy  = —^, 
<f>{xy) 

whence 

4>(xy)d{xy)  =  dz. 


I 


§  XX III.]  EXAMPLES   OF  MONGERS  METHOD.  337 

Since  <^  is  arbitrary,  the  integral  of  the  first  member  is  an  arbi- 
trary function  of  xy,  hence  we  may  write 

z-^{xy)  =(3; (9) 

and  finally  putting  P  =  ^  (a),  we  have 

z  =  <t>(xy)  +  il/{x-\-y), (10) 

which  is  therefore  the  general  integral  of  equation  (i). 

Another  intermediate  integral  might  have  been  found,  but 
less  readily,  by  employing  the  other  factor  of  equation  (2). 

310.  When  either  of  the  variables  z,  p  qx  q  is  contained  in 
R,  S  ov^T,  the  first  of  Monge's  equations  is  integrable  only  in 
connection  with  ds  =  pdz  +  qdy.    For  example,  given  the  equation 

^2^  —  2p^S  -f-  /2/  —  o. 

Monge's  equations  are 

^2^2  -f.  2pqdydx  +  p^dx'^  =  o, 
and 

q^dpdy  +  p^dgdx  —  o. 

The  first  is  a  perfect  square  and  gives  only 

qdy  +  pdx  =  o, 

which  converts  the  second  into 

qdp  —  pdq  =  o. 
Hence  the  integrals 

z  =  a,  and  /  =  <^^> 

and  the  intermediate  integral 

P^q4>{z). 

For  this  Lagrange's  equations  are 


338  EQUATIONS   OF  THE  SECOND    ORDER.  [Art.  3IO. 

,         —dydz 
<t>(z)       o 
whence  the  integrals 

js  =  a,  and  X(f>(a) -i- y  =  p ; 

and,  putting  )8  =  \l/{a),  the  final  integral 

y  +  x<li(z)  =  xl/{z). 

In  this  example  but  one  intermediate  integral  can  be  found ; 
the  form  of  the  final  equation  is  that  considered  in  Art.  301. 

311.  In  the  following  example,  the  second  of  Monge's  equa- 
tions must  be  combined  with  ds  =  pdx  -f-  qdy.     Given 

4/  ,  >. 

r—i= i — : (i) 

X  +y'  ^  ^ 

for  which  Monge's  equations  are 

dy^  —  dx^  =  0, (2) 

Ap 
dpdy  -  dqdx  + -^^j^dydx  =  o (3) 

Taking  from  equation  (2) 

dy  —  dx  =  Oy 

whence  the  integral 

y=^x  +  a, (4) 

equation  (3)  becomes 

^pdx 
dp—  dq  +  -^^-1 —  =  o, 
^         ^    '   2x  +  a        * 

or 

(^2x  -\-d){dp-  dq)  +  A,pdx  =  o (5) 

To  ascertain  whether  this  is  an  exact  equation,  subtract  from 
the  first  member  the  differential  of  {2x  -\-  d)  (p  —  q),  which  is 

{2x  -\-  a)  {dp  —  dq)  -f  2pdx  —  2qdx. 


I 
I 


§  XXIII.]  EXAMPLES   OF  MONGERS  METHOD.  339 

The  remainder  is 

2pdx  +  2qdx, 

which,  since  dx  =  dy,  is  equivalent  to  2d2.     Hence,  equation  (5) 
is  exact,  and  gives  the  integral 

{2x  +  a){p-q)  +  2Z=^b (6) 

From  equations  (4)  and  (6)  we  have  the  intermediate  integral 

{x-\-y){p-q)  +  2Z^<^{y-x) (7) 

Lagrange's  equations  now  are 

dx    _  dy     _  dz 

x-\-y  X -{- y      (ji{y  —  x)  —  2Z 

whence  we  have  the  integral 

x+y  =  a, (8) 

which  converts  the  relation  between  dy  and  dz  into 

dz        2S  _        <i>{2y  —  a) 
dy        a  a 

The  integral  of  this  last  equation  is 

2^     "  =  --J^    ''  <i>{2y-a)dy  +  p..     ....  (9) 

Finally  using  equation  (8)  and  putting  /8  =  -«/'(a),  we  have 

a 

__2y_  -   _2y 

{x-\-y)ze  -^y  =  -\e    <^  <i>{2y-a)dy-\-xl;{x+y),     .     (10) 


:    where  ;r  +  j  is  to  be  put  for  a  after  the  indicated  integration. 

312.  In  this  example  it  was  not  possible  to  obtain  the  second 
integral  required  in  Lagrange's  process  in  a  form  containing  a 
simple  arbitrary  function  of  the  form  <^(z^),  as  was  done  in  finding 
equation  (9),  Art.  309.     Thus  the  final  integral  in  the  present 


340  EQUATIONS   OF  THE  SECOND   ORDER.  [Art.  3 1 2. 

case  is  not  of  the  form  considered  in  Art.  300.  In  the  case  of 
a  primitive  of  the  present  kind,  there  is  but  one  intermediate 
integral.  Accordingly,  it  will  be  found  that,  had  we  employed 
the  other  factor  of  equation  (2),  the  resulting  system  of  Monge's 
equations  would  not  have  been  integrable. 

Examples   XXIII. 
Solve  the  following  partial  differential  equations  :  — 
I.    r^f{x,y),  z  ^\[f{x,y)dx^^x^{y)  +  x\,{y) 

z  =  ^x^  logy  +  axy  +  <^(^)  +  xl/{y) 

z=y{ey  —  e^)  +  <i>{x)  +  eyxlf{x) 

z  =  ^x^'y  —  xy  +  cf>(y)  +  e-^i{/(y) 

z  =  ^x^y  +  <f>(y)  log;(f  +  if/{y) 

z^  =  x^y^  +  x<l>(y)  -\-  ilf{y) 

z  =  log  le^ycf>{y)  -  e-^y'\  +  y\,{y) 

z  =  <f>{x)il/{y)x^°sy 

z—{x  +y)  logy  +  <^{x)  +  i/'(^  +  y) 
x=ct>{z)  +x(,{y) 

11.  x^r -h  2xys -\r y^^  =  o,  ^  =^  "^^^ ( ^ )  + '^ (^j 

12.  r  —  a^t—o,  z  —  f^{y  ■\- ax)  -\- \^{y  —  ax) 

13.  ^V  -  y-'t  =qy-px,  z  =  <f>  I -J  -\-  if;{xy) 

14.  ^(i +^)r-(/  +  ^  +  2/^)j-+/(i  +/)/=o, 

xz=  ct>{z)  +  i(/{x  -^y  +  z) 

15.  (3  +  ^^)^r  —  2(^  -f-  ^g)  {a  +  c/>)s  +  {a-h  cpyt  =  o, 
y  -\-  x<l>{ax  +  3y  -\-  cz)  =  ^{ax  +  dy  +  cz) 


2. 

3- 

t-q  =  e^  J^ey, 

4- 

p  +  r^xy, 

5- 

xr  +  p=.  xy, 

6. 

zr+p-=7,xy^, 

7- 

r+P'=y^ 

8. 

z^ 

9- 

c. 

ps  —  qr  =  0, 

§  XXIV.]  LINEAR  EQUATIONS.  34 j 


XXIV. 

Linear  Equatioiis. 

313.  A  partial  differential  equation  which  is  linear  with  re- 
spect to  the  independent  variable  z  and  its  derivatives  may  be 
written  in  the  symbolic  form 

F{D,D')z=V, (i) 

where 

dx  dy 

and  F  is  a  function  of  x  and  y.  We  have  occasion  to  consider 
solutions  only  in  the  form 

z^f{x,y), 

and  shall  therefore  speak  of  a  value  of  z  which  satisfies  equa- 
tion (i)  as  an  integral.  Since  the  result  of  operating  with 
F{p,  U)  upon  the  sum  of  several  functions  of  x  and  y  is  obvi- 
ously the  sum  of  the  results  of  operating  upon  the  functions 
separately,  the  sum  of  a  particular  integral  of  equation  (i)  and 
the  most  general  integral  of 

F{D,D^)z  =  o (2) 

will  constitute  the  general  integral  of  equation  (i).  Hence,  as 
in  the  case  of  ordinary  differential  equations,  the  general  in- 
tegral of  equation  (2)  is  called  the  complementary  function  for 
equation  (i). 

So  also,  as  in  the  case  of  ordinary  differential  equations,  when 
the  second  member  is  zero,  the  product  of  an  integral  and  an 
arbitrary  constant  is  also  an  integral ;  but  this  does  not,  as  in 
the  former  case,  lead  to  a  term  of  the  general  integral,  since 


342  HOMOGENEOUS  LINEAR  EQUATIONS.     .       [Art.  3 1 3. 

such  a  term  should  contain  an  arbitrary  function.  It  is,  in  fact, 
only  in  special  cases  that  the  general  integral  consists  of  sepa- 
rate terms  involving  arbitrary  functions. 

Homogeneous  Equations  with  Constant  Coefficients, 

314.   The  simplest  case  is  that  in  which  the  equation  is  of 
the  form 

Ao- V  A^- +  .  .  .  +  ^„-—  =  o,   .     .     .  (i) 

doc^  dx^'-'^dy  dy» 

the  derivatives  contained  being  all  of  the  same  order,  and  their 
coefficients  being  constants.     Let  us  assume 

2  =  ^  (jv  +  mx) . 

Now  — -  xpiy  -f-  mx)  =  m\l;'{y  +  mx)  and  — -  ^{y  +  mx)=\l/'(y  +  mx), 
ax  ay 

whatever  be  the  form  of  the  function  xjj,   therefore  the   result 

of  substitution,  after  rejecting  the  common  factor  <^^*^^{y'{-mx)y 

will  be 

Aofn*'  +  A^m^-''  +  .  .  .  +  An  —  o (2) 

Hence,  if  w  be  a  root  of  this  equation,  3  =  <^(j  -h  mx)  satisfies 
equation  (i),  <^  being  an  arbitrary  function.  If  m^,  m^,  .  .  .,  ^n^ 
are  distinct  roots  of  equation  (2),  we  have  the  general  integral 

z=^<f>x{y  +  m^x)  -+-  <fi:,{y  +  m^)  +  .  .  .  +  <iin{y  +  ntnx),  .  (3) 

where  <t>i,  <^2,  .  .  .,  <^«  are  arbitrary  functions. 
Given,  for  example,  the  equation 

d^z  d'^z    ,       ,  d^z 

3a  — —  -f  2^2  — -  =  o. 

dx"^  dxdy  dy^ 

The  equation  for  m  is 

m^  —  ^am  -f-  2^=  =  o. 


§  XXIV.]  WITH  CONSTANT   COEFFICIENTS.  343 

whence  m=za  or  m  =  2a.     Hence  the  general  integral  is 
z=  <f>{y -\-ax) +\l/{y -\- 2ax). 

315.    Equation  (i)  of   the  preceding  article,  when  written 
symbolically,  is 

{AoD^  +  A^D^-^D'  +  .  .  .  +  A„D^^)z  =  o, 

or,  resolving  into  symbolic  factors, 

(Z)  -  m^D^){D  -  m^D')  .  .  .  {D  -  mnD')z  =  o.     .     .  (4) 

Since  the  factors  are  commutative,  this  equation  is  evidently 
satisfied  by  the  integrals  of  the  several  equations, 

{D-m,D')z  =  o,    {JD  -  m^D^)z  =  o,    ...    {D  -  mnD')z  =  o. 

Accordingly  the  several  terms  of  the  general  integral  (3)  are 
the  integrals  of  these  separate  equations. 
Again,  the  equation  may  be  written 

^'"/(|,)-=°.  • (5) 

where/ is  an  algebraic  function  of  the  nth.  degree,  and  equation 
(2)  is  equivalent  to 

f{m)  =  o._ 

We  may  now  regard  the  symbol  — ,  when  operating  upon  a  func- 
tion of  the  form  <^(j/  +  mx)  as  equivalent  to  the  multiplier  m, 
thus 


It  follows  that 


j^My  +  ^^)  =  ^<^(i'  +  ^^)' 


-^\D' '  *^^^  ^  ^^^  =/(^)<f'(y  +  ^^)  > 


344  LINEAR  EQUATIONS.  [Art.  3 1 5. 

SO  that  equation  (5)  is  satisfied  by  ^{y  -f-  mx)  when  f(m)  —  o, 
whatever  be  the  form  of  the  function  <^. 

316.   The  solution  of  the  component  equations,  of  which  the 
form  is 

{D-  jnD')z  =  o (i) 

may  be  symbolically  derived  from  that  of  the  corresponding 
case  of  ordinary  differential  equations.  For,  if  we  regard  D'  in 
equation  (i)  as  constant,  its  integral  is 

where  C  is  a  constant  of  integration.  Replacing  C  by  <^(/),  as 
usual  in  integrating  with  respect  to  one  variable  only,  we  have 
for  the  symbolic  solution 


—  ^mxD' 


Hy), (2) 


where  <^(j)  is  written  after  the  symbol  because  D'  operates 
upon  it,  though  it  does  not  operate  upon  x.  The  symbol  e""^^' 
is  to  be  interpreted  exactly  as  if  D'  were  an  algebraic  quantity. 
Thus 

=  4,{y)  +  mx^\y)  +^^<'(y)  +  .  . ., 

2  ! 

or 

e**"^^'€fi{y)  —  (^{y  -\-  mx), 

by  Taylor's  theorem,  of  which  this  is  in  fact  the  symbolic  state- 
ment (Diff.  Calc.  Art.  176). 

It  should  be  noticed  that  the  process  of  verifying  the  identity 

{D  -  mD')e^^D'<i>{y)  =  o, 


§  XXIV.]  CASE    OF  EQUAL  ROOTS.  345 

with  the  expanded  form  of  the  symbol  e*"^^',  is  precisely  the 
same  as  that  of  verifying 

{D  —  m)e"^^  =  o, 
with  the  expanded  form  of  the  exponential  e"^"^. 

Case  of  Equal  Roots. 

317.  The  general  solution,  equation  (3),  Art.  314,  contains 
n  arbitrary  functions ;  but  when  two  of  the  roots  oif{m)  =  o  are 
equal,  say  m^^  =  niz,  the  corresponding  terms, 

are  equivalent  to  a  single  arbitrary  function.  There  is,  how- 
ever, in  this  case  also,  a  general  integral  containing  n  arbitrary 
functions.     To  obtain  it  we  need  an  integral  of 

{Z>-m,D')'z=o,    .......  (i) 

in  addition  to  that  which  also  satisfies  (B  —  m^U)z  =  o.  This 
required  integral  will  be  the  solution  of 

{D  -  m^D^)z  ^  <^{y -^  m^x)  ; (2) 

for,  if  we  operate  with  D  —  m^D^  on  both  members  of  this  equa- 
tion, we  obtain  equation  (i),  so  that  its  integral  is  also  an  in- 
tegral of  equation  (i). 

Writing  equation  (2)  in  the  form 

p —  m^q=  ^{y-\-m^x), 

Lagrange's  equations  are 

^^=_!Si= ^ , 


346  LINEAR  EQUATIONS.  [Art.  3 1 7. 

of  which  the  first  gives  the  integral 

y  -j-  tn^x  =  a, 

and  then  the  relation  between  dx  and  dz  gives 

z  =  x<l>{a)  +  3. 

Thus  the  integral  of  equation  (2)  is 

z  —  x<f>{y  +  m^x)  -f  il/{y  +  m^x)  ; 

and,  regarding  <^  and  «A  as  both  arbitrary,  this  is  the  general  inte- 
gral of  equation  (i). 

318.    The  solution  may  also  be  derived  symbolically ;  for, 
since  the  solution  of 

(D  —  nifz  —  o 
is 

we  have,  for  the  solution  of 

(Z>  —  mDyz  =  0, 

z^e^^i>'\_xf^{y)^-y\,{y)\ 
that  is, 

z^=  x^^i^y -Y  mx) -^-y^i^y -\- mx) (1) 

The  solution  might  also  have  been  found  in  the  form 

z=y^x{y -^mx)-\-\\i^{y-\rnix), (2) 

but  this  is  equivalent  to  the  preceding  result ;  for  we  may  write 
it  in  the  form 

^  =-\y  ^-nix  —  mx)<f>i(y  +  mx)  -h  «/'i(>'  +  mx)  ; 
and,  since  (jy  +  mx)  <t>i(jy  +  mx)  -\-^i(y  +  mx)  and  —  m4>i{y  4-  mx) 


§  XXIV.]  CASE   OF  IMAGINARY  ROOTS.  347 

are  two  independent  arbitrary  functions  of  y-\-mx,  they  may 
be  represented  by  \p  and  <^,  the  equation  thus  becoming  identical 
with  equation  (i). 

In  like  manner,  if  the  equation  /( —  J  =  o  has  r  equal  roots, 
the  terms  corresponding  to  {D  —  mUy  are 


Case  of  Imaginary  Roots, 

319.   When  the  equation  has  a  pair  of  imaginary  roots,  /t  ±  ivy 
the  corresponding  terms  in  the  general  integral  are 

z  =  <l>(y  +  fjLx  -\-  ivx)  +  \p{y  +  fix  —  ivx)  ; 

or,  putting  u=zy-\-fix,  v  —  vx, 

<^(«  +  iv)  +  «/'(«  —  iv). 

To  reduce  this  expression  to  a  real  form,  assume 

so  that 

<^  =  i(<Ai  +  ^«/'i)>  and  ^  =  ^  (<t>i  -  i^i) * 

Making  the  substitutions,  the  expression  becomes 

2  =  i-[<^i(«  +  t'y)  +  <kx{u  -  iv)']  +^[«Ai(«  +  iv)  -  ^i(u  -  tv)^. 

In  this  expression  <^i  and  if/j  are  arbitrary  functions,  since  <^  and 
ip  were  arbitrary ;  but  giving  any  real  forms  to  <^i  and  lAxjthe  two 
terms  are  real  functions  of  ti  and  v,  that  is  to  say,  real  functions 
of  X  and  y. 


348  LINEAR  EQUATIONS.  [Art.  3 1 9. 

Given,  for  example,  the  equation 

=  o, 

dx^      dy^         ' 

of  which  the  solution  in  the  general  form  is 

z  =  ^{x-\-iy) +\^{x  —  iy). 
In  the  form  given  above  the  solution  is 

If,  for  instance,  we  assume  cf)^{t)  —  P  and  i/'i  (f)  =  e^,  we  have  the 
particular  solution  in  real  form 

z  =  x^  —  ^xy^  +  <?-^  sin>', 
which  is  readily  verified. 

The  Particular  Integral, 

320.  The  methods  explained  in  the  preceding  articles  enable 
us  to  find  the  complementary  function  for  an  equation  of  the 
form 

F(^D,D^)z^V, 

when  F{p^  D^)  is  a  homogeneous  function  of  D  and  D\  and  Fa 
function  of  x  and  y.  The  particular  integral,  which  is  denoted 
by 

I V 

can  also  in  this  case  be  readily  found. 

Resolving  the  homogeneous  symbol  F{Df  D')  into  factors,  we 
may  write 

F(D,  £>')  =  (Z)  -  m,D')  {D  -  m^)  .  .  .  {D  -  ninD'), 


§  XXIV.]  THE  PARTICULAR   INTEGRAL.  349 

and  the   inverse  symbol  may  be  separated,  as  in  Art. 

105,  into  partial  fractions  of  the  form 

where  the  numerators  are  numerical  quantities,  and  ;'  is  unity 
except  when  multiple  roots  occur.  It  is  therefore  only  neces- 
sary to  interpret  the  symbol 

321.    For  this  purpose  we  employ  the  formula 

proved  in  Art.  116.  Putting  mD'  in  place  of  a,*  this  formula 
gives 

=■  e^^^' —  ^{x,  y  —  mx) (i) 

Hence  the  result  is  found  by  subtracting  mx  from  j^  in  the  oper- 
and, integrating  with  respect  to  x,  and  adding  mx  to  y  after  the 
integration.     Since 

*  In  explanation  of  this  application  of  the  symbolic  method,  let  it  be  noticed 
that,  just  as  the  formula  of  Art.  116  is  founded  upon  the  equation 

De^^  V^  e^^{D  +  a)  V, 

so  the  present  application  of  it  depends  upon 

De*"^^'^{x,  y)  =  e^^'^XO  +  wZ)')  ^{x,  y), 
or 

Di(x,  y  -h  mx)  =  result  of  putting  y  +  mx  for  _y  in  (Z>  +  mD')  ^(x,  y)y 

which  expresses  an  obvious  truth. 


350  LINEAR  EQUATIONS.  [Art.  32 1. 

j'^(^,  y  -  mx)dx  =  f^H,  y  -  mi)d^, 
this  may  be  expressed  by  the  equation 

£>  -mD'  ^^^'  ^^  ^  r*^^^'  y  +  mx-  mi)di.     ...  (2) 

In  like  manner,  for  the  terms  corresponding  to  multiple  roots 
olf{m)  =  o,  we  have 


(^ 


;^;^,^(^,  J^)  =  j"J.  .  >^^{k,y^mx  -  mi)di^,     .  (3) 


322.  There  are  certain  methods  by  which,  in  the  case  of 
special  forms  of  the  operand,  the  result  may  be  obtained  more 
expeditiously  than  by  the  general  method  just  given.  Some  of 
these,  which  apply  as  well  when  the  equation  is  not  homogene- 
ous, will  be  found  in  Arts.  328-334.  The  following  applies  only 
when  the  equation  is  homogeneous. 

Suppose  the  seco7id  member  to  be  of  the  form,  ^{ax  +  by).  The 
equation  may  be  written  in  the  form 

F{D,  D')z  =  ^y f^') 2  =  ^(ax  +  by). 
It  is  readily  seen  that 

f(^§^^(^^  +  by)  =/g)  H^^  +  by). 
We  have,  therefore,  for  the  particular  integral 

or,  denoting  ax  +  by  by  t,  since  a"f[-]  =-^{^>  ^)> 


§  XXIV.]      SECOND  MEMBER   OF  THE  FORM  ^ {ax -\- by).      35  I 

^=^fl- ••!*«'''"•• ^^> 

Given,  for  example,  the  equation 

2  —  =  sm  ix  +  2v) , 

dx^      dxdy  dy^  ^  -^^' 


the  particular  integral  is 


—  sin  {x  +  2^) 


X)2  _|_  2)/)'  _  2Z>'^ 

= ^- sin  tdt^  =  i  sin/  =  i  sin  (;t:  +  2^). 

1  +  2  —  8JJ  5  5 

Adding  the  complementary  function, 

z  =  (^(^y  4-  ^)  +  ^{y  —  2^)  +  ^  sin  {x  +  2y), 

323.  When  i^(<a:,  b)  =  o,  the  operand  is  of  the  form  of  one  of 
the  terms  of  the  complementary  function.  The  method  then 
fails,  the  expression  given  in  the  preceding  article  representing 
a  term  included  in  the  complementary  function,  with  an  infin- 
ite coefficient.  In  this  case,  after  applying  the  method  to  all 
the  factors  of  the  operative  symbol,  except  that  which  vanishes 
when  we  put  D  =  a  and  D'=  b,  the  solution  may  be  completed 
by  means  of  the  formula 

D-mD''^^^  4-  mx)  =  xf{y  +  mx), 
which  results  immediately  from  equation  (i),  Art.  321. 


*  This  integral  involves  an  expression  of  the  form  At^~^  +  Bt"~^  +  .  .  .  +  Z, 
in  which  A,  B,  .  .  .,L  are  arbitrary  constants,  but  such  an  expression  is  included  in 
the  complementary  function.  It  must  be  remembered  that  the  multiple  integral  in 
equation  (i)  is  not  to  be  regarded  as  involving  an  arbitrary  function  oiy. 


352  LINEAR  EQUATIONS.  [Art.  323. 

Thus,  if  in  the  example  given  in  the  preceding  article  the 
second  member  had  been /(;r +_;/),  we  should  have  had 


D{D  -D')  D+  2D 


=  ^YAx+y)dx. 


The  Non- Homogeneous  Equation, 

324.   When  the  equation 

F{D,D^)z=zo (i) 

is  not  homogeneous  with  respect  to  D  and  D\  the  solution  can- 
not generally  be  expressed  in  a  form  involving  arbitrary  func- 
tions.    Let  us,  however,  assume 

z  =  ce^^  +  ^y, (2) 

where  Cy  h  and  k  are  constants.  Substituting  in  equation  (i), 
we  have,  since  Z>^*^+^  =  he^^'+^y  and  He^^'-^^y  =  ke^^+*y, 

cF{h,  k)e^^-^f'y=zo. 

Thus  we  have  a  solution  of  the  assumed  form,  if  h  and  k  satisfy 
the  relation 

J'{h,k)^o, (3) 

c  being  arbitrary.     Let  equation  (3)  be  solved  for  h  in  terms  of 

k.     Now  if  F{hy  k\  is  homogeneous,  we  shall  have  roots  of  the 

form 

h  =  m^kf      h  =  m^ky      .  .  .,      h  =  m„k ; 


§  XXIV.]  THE  NON-HOMOGENEOUS  EQUATION.  353 

and,  since  the  sum  of  any  number  of  terms  of  the  form  (2)  which 
satisfy  the  condition  (3)  is  also  a  solution,  the  equation  will  be 
satisfied  by  any  expression  of  the  form 

where  m  has  any  one  of  the  values  m^,  m^y  .  .  .,  m„.  But,  since 
for  a  given  value  of  m  this  expression  is  a  series  of  powers  of 
^y+mx  with  arbitrary  coefficients  and  exponents,  it  is  equivalent 
to  an  arbitrary  function  of  ^>'+'«-^,  that  is  to  say,  it  denotes  an 
arbitrary  function  oi  y  -\-  mx.  This  agrees  with  the  result  other- 
wise found  in  Art.  314. 

325.  Again,  if  F{D,  U)  can  be  resolved  into  factors,  and 
one  of  these  is  of  the  form  D  —  mH  —  by  so  that  F{ky  ^)  =  o  is 
satisfied  by 

h  =  mk  +  b, 

equation  (i)  will  be  satisfied  by  an  expression  of  the  form 

z  =  %ce^^y + '«-^) + ^^, 

where  m  and  b  are  fixed  and  c  and  k  are  arbitrary.  But  this  ex- 
pression is  equivalent  to  the  product  of  ^^-^^  into  an  arbitrary 
function  of  7  +  mx.  Thus,  corresponding  to  every  factor  of  the 
form  D  —  mD  —  ^  we  have  a  solution  of  the  form 

z  =  e^^<^{y  +  mx). 

Given,  for  example,  the  equation 

d^z  _  d^   I   ^       ^  _- 
dx"^      dy"^       dx      dy         * 
or 

the  general  integral  is 


354  LINEAR  EQUATIONS.  [Art.  325. 

We  might  also  have  found  the  solution  in  the  fori 

z=^  ^,{y  -  X)  ^  eyy\,^{y  -^  X)  ', 

but,  writing  the  last   term  in  the  form  ey^^~*\\i^{y  -\-  x)^  this 
agrees  with  the  previous  result  if  «/'(/)  is  put  for  e^\p^{t). 

326.  In  the  general  case,  however,  we  can  only  express  the 

solution  of 

F{D,D')z  =  o (i) 

in  the  form 

z  =  ^ce'^^  +  ^y, (2) 

where 

F{h,  k)  =  o, (3) 

so  that  c  and  one  of  the  two  quantities  h  and  k  admit  of  an 
infinite  variety  of  arbitrary  values. 
Given,  for  example,  the  equation 

d^z      dz  _ 
dx^       dy 

Here  F{p,  U)  =  Z>'  —  D\  whence  k^  —  k  —  o,  thus  the  general 

integral  is 

z  ==  '^ce''^^  +  ^'^y. 

Putting  h-=  I,  h  —  2,  k  =  \,  etc.,  we  have  the  particular  integrals 
e^+y^  e'^-^^y  ^i^+b',  etc. 

Special  Forms  of  the  Integral, 

327.  There  are  certain  forms  of  the  integral  of  F(Z>,  D^)z  =  o 
which  can  only  be  regarded  as  included  in  the  general  expres- 
sion (2),  by  supposing  two  or  more  of  the  exponentials  to  become 
identical.     Let  the  value  of  k  derived  from  F{liy  /^)  =  o  be 

k^Ah), (4) 

then 

hi  —  ^2 


§  XXIV.]  SPECIAL  FORM^    OF   THE  INTEGRAL.  355 

is  an  integral  of  FiDy D')z  =  o.    When  h^^h^-Bs. h^  this  takes  the 
indeterminate  form,  and  its  value  is 

dh 
which  is  accordingly  an  integral.     In  like  manner  we  can  show 

that  —  ^>^-^+/(^):»',  and  in  general,  — 1-^A-^+/(A):»'  satisfies  equation 
dk'  dk-  ^ 

(i) ;  thus  we  have  the  series  of  integrals 


.(5) 


For  example,  in  the  case  of  the  equation  (D^  —  D');s  =  o,  the 
integral  ,?'^-*+^'y  gives  rise  to  the  integrals 

e^^  +  ^'y  (x-^  2hy), 

/■^  +  ^''^[(^  4-  2kyy  +  6y(x  +  2kyy], 

e^^  +  ^'>'[{x  +  2ky)^  +  i2y{x  +  2/iy)^  +  J2y^2f 

In  particular,  putting  /i  =  o,  we  have  the  algebraic  integral 
z  —  CiX-\-  c^^x"^  4-  2y)  +  c^{x'i  -\-  6xy) 

Special  Methods  for  the  Particular  Integral. 
328.  The  particular  integral  of  the  equation 


356  LINEAR  EQUATIONS.  [Art.  328. 

is  readily  found  in  the  case  of  certain  special  forms  of  the  func- 
tion V. 

In  the  first  place,  suppose  V  to  be  of  the  form  e^^-^^y.  Since 
De^^^h^ae^'^'^^y  and  D'c^'^^^y  ^bc^'^'^^y,  and  FiDy  D')  consists 
of  terms  of  the  form  D'^U^  we  have 

F(^D,  Z>')  ^-^  +  ^^  =  F{a,  b)  ^«^  +  h^ 
or 


^ F(a,  b)e''^  +  ^y  =  e^^  +  h 


where  Fifty  b)  is  a  constant.     Hence,  except  when  F{ay  b)  =  o, 
we  have 


.  ^ax  +  iy  ^z  ^-^  +  ^y. 


F{Dy  D^)  F{a,  b) 

Thus,  when  the  operand  is  of  the  form  e'^^'^^y^  we  may  put  a  for 
D  and  b  for  D\  except  when  the  result  introduces  an  infinite  co- 
efficient.    Given,  for  example,  the  equation 

the  particular  integral  is 

Z  =  ^ ^2j:  +  J/  _   1  ^2jr  +  >, 

329.  In  the  exceptional  case  when  F{a,  b)  =  o,  we  may  pro- 
ceed as  in  Art.  no.  Thus,  first  changing  a  in  the  operand  to 
a  +  h,  we  have 

Z= ^x+Ax+iy^  1 gax+3yfj^/ix^^^^  _\ 

F{D,F>')  F{a-\-h,b)  \  2  ) 

The  first  term  of  this  development  is  included  in  the  comple- 
mentary function.  Omitting  it,  we  may  therefore  write  for  the 
particular  integral 


§  XXIV.]  SECOND  MEMBER    OF  THE  FORM  e^^-^h,  357 


{x-\-\hx-^  -^  .  .  .)e^^  +  ^y, 


F{a  +  A,d) 


in  which  the  coefficient  takes  the  indeterminate  form  when  /i  =  o, 

because   F(a,d)=o,   and  its  value  is  — — — -,  where  FJia,  b) 

^a\(i,  0) 

denotes  the  derivative  of  Fia,  b)   with  respect  to  a.      Hence, 
except  when  Fa  {a,  b)  =  o,  we  have 

e'^^  +  h (i) 


FJ{a,  b) 


In  Uke  manner,  if  Faia,  b)  =  o,  the  second  term  of  the  de- 
velopment is  in  the  complementary  function,  and  we  proceed  to 
the  third  term.  It  is  evident  that  we  might  also  have  obtained 
the  particular  integral  when  F{ay  <5)  =  o  in  the  form 

eax  +  by.^   ........  (2) 


Fi\a,  b) 
but  the  two  results  agree,  for  their  difference, 


J_1 


_FJ{a,  b)       F,\a,  b)] 


gax  +  by^ 


is  readily  seen  to  be  included  in  the  first  of  the  special  forms 
(5)  of  Art.  327,  since  a  and  b  are  admissible  values  of  the  h  and 
k  of  that  article. 

330.  In  the  next  place,  let  V  be  of  the  form  sin  {ax  +  by)  or 
cos  {ax-\-by).  We  may  proceed  as  in  Arts.  11 1  and  112,  and  it 
is  to  be  noticed  that  we  have,  for  these  forms  of  the  operand,  not 
only  V^  —  a""  and  n""  =  —  b"",  but  also  DD^  =  —  ab.  Given,  for 
example,  the  equation 

d'^z   .     d^^z     ,   dz  •    /      ,       \ 


35^  LINEAR  EQUATIONS.  [Art.  330. 


the  particular  integral  is 


z  — 


sin  {x  +  2y)  —  — ^ —  sin  {x  +  2y) 


D^-\-DD'-^D' -\       '  ^'      Z>'-4 


/)'  +  4 


sin  (jc  +  2y)  =  —  tV  [cos  (^  -h  27)  +  2  sin  {x  +  2jv)] . 


Z>'»-  16 
Adding  the  complementary  function,  we  have 

z  =  e^^i^y  —  x)  -\-  e-^\p{y)  —  j\  cos  {x  +  2j;)  —  -J-  sin  (x  -f  2>'). 

The  anomalous  case  in  which  an  infinite  coefficient  arises 
may  be  treated  like  the  corresponding  case  in  ordinary  differen- 
tial equations. 

331.  Again  /et  V  be  of  the  form  xy%  where  r  and  s  are 
positive  integers.  In  this  case,  we  develop  the  inverse  symbol 
in  ascending  powers  of  D  and  D'.  Thus,  if  the  second  member 
in  the  example  of  the  preceding  article  had  contained  the  term 
x^yy  the  corresponding  part  of  the  particular  integral  would  have 
been  found  as  follows : 


z  == x^y 

i-{D^Jf.  DV  +  D')    ^ 

=  -  [i  +  (^^  +  DD'  +  D')  +  {D^  +  DD*  +  ny  +  .  .  ."Ixy 

=  -  [i  -f-  Z)^  -f  DD'  -f  Z)'  +  2D^D''\  x^y 

St  —  x'^y  —  2y  —  2x  —  X'  —  4. 

It  will  be  noticed  that,  on  account  of  the  form  of  the  operand,  it 
is  unnecessary  to  retain  in  the  development  any  terms  containing 
higher  powers  than  Z>^  and  D'.  Again,  had  the  operand  been 
xy,  we  might  have  rejected  D^  in  the  denominator  thus  : 


^y= 7:^. — —^^y 


-[i4-Z>'(i  +Z>)]^j'=  -xy-x—  I. 


I 


§  XXIV.]  SECOND  MEMBER   OF  THE  FORM  x^f.  359 

332.  When  the  symbol  F{p,  Z?')  contains  no  absolute  term, 
we  expand  the  inverse  symbol  in  ascending  powers  of  either 
D  or  D\  first  dividing  the  denominator  by  the  term  containing 
the  lowest  power  of  the  selected  symbol.  For  example,  given 
the  equation 

dx^         dxdy 

for  the  particular  integral  we  have  to  evaluate 


In  this  case,  it  is  best  to  develop  in  ascending  powers  of  D\ 
because,  with  the  given  operand,  a  higher  power  of  D  than  of 
D^  would  have  to  be  retained.     Thus 


^:p^)--=iC-^3f)-. 


D^  D^  12       20 

Adding  the  complementary  function, 

2  =  <^  W  +  ^{y  +  3^)  +  -hx^y  +  i^x\ 

If  we  develop  the  symbol  in  ascending  powers  of  Z?,  the  par- 
ticular integral  found  will  be 

_      x^y"^      x^y"^       xy^ 
~        18         54        324* 

The  difference  between  the  two  particular  integrals  will   be 
found  to  be 

TiE^li3x+yy-y'2, 

which  is  included  in  the  complementary  function. 


36o  LINEAR  EQUATIONS.  [Art.  333. 

333.    Finally,  whefi  the  operand  is  of  the  form  e^'^^yVy  we 
may  employ  the  formula  of  reduction 

F{D,  D')e^^  +  ^yV=  e^'^hF{D  -\- a,  D'  -\-  b)  V, 

which  is  simply  a  double  application  of  the  formula  of  Art.  116. 
For  example, 


_  gax  +a?y  _J I ^ 

2aD  ^       D^ 

2a 


2a 


2a      D\       2a  J  \^a      ^a^J 

If  we  develop  in  powers  of  D\  we  shall  find 

^ xe^^+'^^y  =  —  e^x  +  '*''y(xy  -\-  ay'). 

The  difference  between  the  two  results  is  accounted  for  by  the 
special  forms  given  in  Art.  327  for  the  complementary  function 
in  this  example. 

334.   As  another  application  of  the  formula,  let  us  solve  the 
equation 

dx*      dxdy         dy 

The  particular  integral  is 


z  =  the  coefficient  of;  in  ___^^_^^^-+^>i^. 


Now 


gix  +  lyx^  —  <f  i-r  +  y ; ^ X^ 


J>  H-  DD^  -  6X>'=^  {D  +  iy  +  i{D  +  /)  -  6;  = 

I 


-^gtx  +  iy. 


D^  +  ZiD  +  A 


§  XXIV.]  SECOND  MEMBER   OF  THE  FORM  e^^'  +  hy.        36 1 

and  by  development  we  find 


therefore 
I 


D^-VDD'-dD'^ 


gix + tyx^  =  [cos  {x +)>)•{'  i  sin  {x  +  j;)]  T—  -  3^  -  ill 

L4        8        32J 


Taking  the  coefficient  of  /,  and   adding  the   complementary 
function, 

2=(  — -  — )sin(^^->')-^cos(:v+>')4-<^(>'+2^)  +  ^/'(J-3^)• 
\4      32/  « 

Linear  Equations  with  Variable  Coefficients. 

335.  In  some  cases  a  linear  equation  with  variable  coeffi- 
cients can  be  reduced,  by  a  change  of  the  independent  variables, 
to  a  form  in  which  the  coefficients  are  constant.  As  an  illus- 
tration, let  us  take  the  equation 

2_  d^ L  ^  —  JL  ^ L  (^  (  \ 

x^  dx^      ■x3  dx      y^  dy^      y"^  dy 


The  first  member  may  be  written  in  the  form 


I  fi  ^^2 i^  dz~\  _\  d      I 

X  \jc  dx^      x^  dxj      X  dx    x 


dz_ 
dx 


Hence,  if  we  put  xdx  =  d^^  whence  I  =  \x'^y  and  in  like  manner 
V  =  \y^i  the  equation  becomes 

dt~  dff ^^^ 

The  integral  of  this  equation  is  ^  =  <3!>(|+ r;)  + 1/^(^— ■>;) ;   hence 
that  of  equation  (i)  is 

z  =  <j>{x^  -iry^)  +  xIj{x^  -r)- 


362  LINEAR  EQUATIONS.  [Art.  336. 

336.  In  particular,  it  is  to  be  noticed  that  an  equation  all  of 
whose  terms  are  of  the  form 

Ax^'y'- — — 

is  reducible  to  the  form  with  constant  coefficients,  like  the  cor- 
responding case  in  ordinary  equations,  Art.  123,  by  the  trans- 
formations ^  =  log  ;»r,  17  =s  log/,  which  give 

dx      di  dy       d-q 

But,  if  we  put  t?  =  X—-  and  i9'  =  j^  — -,  we  may  still  regard  x  and  y 
dx  dy 

as  the  independent  variables ;  the  transformation  is  then  effected 

by  the  formula 

dx^'dy^ 

and  the  equation  reduced  to  the  form 

F{J^,  ^^)z  =  V, 

The  solution  of  this  equation  may  therefore  be  derived  from 
that  of  the  equation  F{Dy  D')  .s"  =  F,  by  replacing  x  and  y  by 
log;ir  and  logy  \  or  it  may,  as  in  the  following  articles,  be  ob- 
tained directly  by  processes  similar  to  those  employed  in  deriv- 
ing the  solution  of  F{D,  D')s  =  V. 

337.  Since 

^x^y  =  rx'^f,  ^^x'^y*  =  sx^y^, 

it  is  obvious  that 

F(^^,^')x^y^  =  F{r,s)x''y.  ......  (i) 


§  XXIV.]  THE  EQUATION  F(J^,  ^')  z  =  O.  363 

Hence,  if  in 

F{^9,  ^')z  =  0, (2) 

we  assume 

z  =  cxy*f 

the  result  is 

{:F{r,  s)x^y^  =  o, 

and  we  have  a  solution  of  the  proposed  form  if  F(ry  s)  =  o. 
Hence  the  general  solution  of  equation  (2)  is 

2  =  2^^0^, (3) 

where 

^(^  ^)  =  o, (4) 

that  is,  2^  is  a  series  in  which  the  coefficients  are  arbitrary, 
and  the  exponents  of  x  and  _>/  are  connected  by  the  single 
relation  (4). 

Now  let  equation  (4)  be  solved  for  r  in  terms  of  s;  if  the 
function  F(^,  ^')  be  homogeneous  in  ^^  and  t?',  the  equation  will, 
have  roots  of  the  form 

r  =  m^s,  r  =  m^s,  etc., 

and  to  each  root  will  correspond  a  solution  of  the  form 

y  —  %c{yx'*'y. 

But  this  represents  an  arbitrary  function  of  yx^.  Thus  to  each 
factor  of  f\^,  ^')  of  the  form 

there  corresponds  an  independent  term  of  the  form 

z  =  4>{yx^) 
in  the  solution  of  equation  (2). 


364  LINEAR  EQUATIONS.  [Art.  337 

Again,  corresponding  to  a  factor  of  the  form 

^  -  m^V  -  b, 

we  have  the  root  r  =  wj  +  by  for  F{ry  s)  =  o\  and  hence  the  solu- 
tion s  =  ^ijx*"yx^y  or 

338.  For  the  particular  integral  of  the  equation 

we  may  suppose  V  to  be  expanded  in  products  of  powers  of  x 
and  J/.     By  equation  (i)  of  the  preceding  article,  we  have 

which  gives  the  particular  integral,  except  when  F{a,  d).=  o. 
When  this  is  the  case,  we  have,  first  putting  a  +  k  in  place  of  a^ 

1 — ^^a+Ay  = 1 ^y(i  ^  klosx  +  .  .  .), 

or,  rejecting  the  first  term  of  the  expansion,  which  is  included  in 
the  complementary  function,  and  then  putting  k  =  o, 

jc«y  =z= x^y^  log  X. 

/^(^'^')     ^       Fa\a,b)      ^      ^ 

339.  As  an  illustration,  let  us  take  the  equation 

x^ V*  —  =  xy. 

dx^      -^  dy^        '^' 

which,  when  reduced  to  the  t^-form,  is 

,^(,?_  1)2  _,:/'(/>'_  i)z  =  xy, 


§  XXIV.]  THE  EQUATION  F(^^,f^^)z=iV.  365 


or 


The  coipplementary  function  is 


for  the  particular  integral. 


I  II 

xy  = xy 


or,  rejecting  the  term  —xy  included  in  ^{xy),  and  putting  ^  =  o, 


2  =  <^{xy)  -\- xy\i[±\ -\-  xy  logjxr. 


^O 


340.  The  symbol  t^  +  '*^'  may  be  particularly  mentioned  on 
account  of  its  relation  to  the  homogeneous  function  of  x  and  y. 
Putting 

we  have  irx'^y^  =  {r-\-s)xy^ ;  hence,  if  u„  denotes  a  homogeneous 
function  of  x  and  y  of  the  nth  degree,  we  have 

7rU„  =  nu„, 

where  u„  is  not  necessarily  an  algebraic  function,  but  may  be  any 


function  of  the  form  x"fii-i     This  is,  in  fact,  the  first  of  Euler's 

theorem  concerning  homogeneous  functions.      See  Diff.  Calc, 
Art.  412. 

As  an  example  of  an  equation  expressible  by  means  of  the 
single  symbol  tt,  let  us  take 


366  LINEAR  EQUATIONS.  [Art.  34O. 

x*'-—  +  nx'*--'y- T  +  — ^-x»-'y'- — 4-...  =  K    (i) 

^x**  dx^-^dy  2  dx^-^dy^ 

The  first  member  can  be  shown  to  be  equivalent  to 

7r(7r  —  l)  .  .  .  (tt  —  «  +  i)z. 

Denoting  this  by  F{Tr)z,  we  have 

F{Tr)u^  —  m{m—i).,.{m^n+i)u^,     .     .     .(2) 

which,  when  F(Tr)  is  expressed  as  in  equation  (i),  is  the  general 
case  of  Euler's  theorem.  Thus  the  complementary  function  for 
equation  (i)  is 

Uo-\-Ur  +  u^  +  ,  .  .-{-  U„-r. 

Let  F  contain  the  given  homogeneous  function  Hf„,  equation  (2) 
gives  for  the  corresponding  term  in  the  particular  integral 


m{m  —  i)  .  .  .  {m  —  n  -\-  i) 


except  when  m  is  an  integer  less  than  ;/.  In  this  case  F{Tr)  will 
contain  the  factor  Tr  —  m,  and  putting  F{ir)  =  {Tr  —  m)<j>(Tr)  we 
readily  obtain  as  in  Art.  338 


Examples  XXIV. 
Solve  the  following  partial  differential  equations  :  — 

d^  z  d^z  d^z 

d^z  d^z     ,  d^z       I 

2. 2 1 = —•) 

dx^dy  dxdy^      dy^      x^ 

'  z  ==  <^(^)  ^if,(x-}-y)  +  xxix  -\-y)  —  j'log^. 


§  XXIV.]  EXAMPLES.  367 


dx^  dxdy  dy^      y  —  2x'     ' 

z  —  <f>{y  —  2x)  +  xp{y  —  3^)  -f.  ^  log  {y  —  ix). 

d'z  d^z     ■      d^z      dz       dz  _ 

dx^  dxdy         dy^       dx      dy 

z  =  <ji{y  +  x)+  e-^xp{y  +  2x). 

d^z      d'^z         dz    ,      dz         ^.  ,«   , 
dx^       dy^  dx  dy 

z  =  ^{x-]-y)-\-e^yi{;(y-x) 

-  ye^+^y  -  (j\x^  +  ixy  +  ix^-hixy+  ^\x) . 

6.   -^  ^a—-{-3—  +  abz  =  e"^^  +«^ 
dxdy         dx         dy 

fitny  +  nx 

z  =  e-^y(f>(x)  +£-^^ily{y)  + 


{m  -\-  a){n  +  b) 


d^z        d^z     ,   dz  /      ,       \    ,     V 

7-    TT  -  -1—7  +  ~r  -  z  =  cos{x  -\-  2y)  +  e^, 
dx^      dxdy      dy 

zz=z  e^ (jii^y)  -{-  e- ■^ ^{x  -\- y)  -\-  ^  ^\n{x  +  2y)  —  xey. 

o     d'^z        d^z     ,   dz       dz  •    /      ,     \ 

8. =  2  sin(^  -\- y)i 

dx^      dxdy      dx      dy  v     ^-^/' 

z  —  e-''<\>{y^-\-\\i{x-{-y^-\-x\%v!\{x-\-y^—  cos  {x-\-y)'\. 

dz  dz         ^^ 

9. a  —  =  e**^"^  cos  ny, 

dx  dy 

z=.  rf)  (  y  +  ax)  H (m  cos  ny  —  na  sin  m^) . 

m^  +  n'^a^^ 

d'^z      d^z   ,   dz    .       dz  ^    «         „ 

dx^       dy""       dx  dy 

z  =  e^4>{y  —  x)-\-  e-^'^^if/  {y  +  x) 

-ie^-y  +  ixy-\-^x-  +  4^xy  +  ix  +  y  +  ^. 


368  LINEAR  EQUATIONS.  [Art.  340. 

II.    mn (;//'  +  w=) +  mn——-\- mn^ tn^n — 

dx"^  dxdy  dy^  dx  dy 

=  cos  {kx  +  /f)  +  cos  {mx  +  ny), 
zss<f>(ny  +  mx)  +  e-»^ij/{my  +  «^) 

ffl;gsin(^j[:  +  i^)  — (ffl/^  —  ;g/)cos(^.y +  /v) 
(;2/&  —  m/)  [m^n^  +  {mk  —  niy] 

mnx  cos{mx  +  ny)  +  {m'^  —  ;g').y  sin(;^;c  +  ny) 

,^«2  ,  ^=*2    ,     ^d^z  dz  dz    , 

13.  x'-r-  -\'2xy——---^y^- nx- ny—-  +  nz  =  o, 

dx''  dxdy         dy^  dx  dy 

d^  z  dz  f 

14.  (^+^)^-^^  =  °'  2  =  \(x+yy'<t>(^)dx  +  ^{y)' 

d^z  ,  d^z        v-xf  \ 

16.   Derive  the  particular  integral  of 


d^z  ,   d^z        ^_xf        ,        \ 


in  the  form  z  =  ^_>'<f^' 


I 


P' 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

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Re^wfid  l^cwks  ateswbjtctfto  r^l4ied|itc  recaiL  6 


9Jun^60V" 


REC'D  LD 


MAY  27  I960 


lygCfTft'tK 


APR  8  -  1956     5 


NIAR2  9  195S14 


OCT  17  1972  3 


LD  21A-50to-4,'60 
(A9562sl0)476B 


General  Libruy 

Unlvenity  of  California 

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